Having changed over the years, the names given to various parts of the lathe are still not completely standardised. No doubt when the government has finished organising every other aspect of our lives it will appoint a highly-paid commission to look into the matter and make "recommendations".
If you would like to buy a book or CD to extend your knowledge of lathes - and how to operate them - look here.
Illustrations of the parts discussed can be found by following the various hyperlinks and also at the bottom of this page. It may be that an Instruction Manual, an Illustrated Parts Book or informative sales Literature may be available for your lathe: you can check by clicking here.
It's still sometimes possible to see advertisements that refer to certain capabilities:
"SS" - sliding and surfacing i.e. with a power feed that slides the carriage along the bed and the cross slide across it.
"BG" - BackGeared for slow speeds  (see below)
"BGSC" - BackGeared and screwcutting
The capacity of a lathe is expressed as its "swing", "throw" "centre height" and "distance between centres"
If you measure from a centre in the tailstock to the nearest point on the bed that is the "centre height", say 5 inches. In the UK this is called the "throw" - or more commonly now as the "centre height" - but doubling it is the "swing", the latter term being used in the USA. In the UK, a 5-inch centre height lathe would be called a "10 inch" in the USA.
The one common measurement is the capacity between centres i.e. the distance between a centre in the headstock and one in the tailstock. The tailstock is moved as far away from the headstock possible but still fully engaged with the bed. It's often possible to gain a few extra inches by having the tailstock overhanging the end of the bed and on light jobs this is acceptable.

BED
The bed of the lathe provides the foundation for the whole machine and holds the headstock, tailstock and carriage in alignment. The surfaces of the bed that are finely machined - and upon which the carriage and tailstock slide - are known as "ways".
Some beds have a gap near the headstock to allow extra-large diameters to be turned. Sometimes the gap is formed by the machined ways stopping short of the headstock, sometimes by a piece of bed that can be unbolted, removed--and lost.
Some very large lathes have a "sliding bed" where the upper part, on which the carriage and tailstock sit, can be slid along a separate lower part - and so make the gap correspondingly larger or smaller.

SADDLE
The casting that fits onto the top of the bed and slides along it is known, almost universally, as the "Saddle" - a self-explanatory and very suitable term.

APRON
The vertical, often flat and rectangular "plate" fastened to the front of the "Saddle" and hanging down in front of itm is known as the "Apron" and carries a selection of gears and controls that allow the carriage to be driven (by hand or power) up and down the bed. The mechanism inside can also engage the screwcutting feed and various powered tool feeds, should they be fitted. The leadscrew, and sometimes a power shaft as well, are often arranged to pass through the apron and provide it with a drive for the various functions. The sophistication of the apron-mounted controls, and their ease of use, is a reliable indicator of the quality of a lathe. Virtually all screw-cutting lathes have what is commonly-called a "half-nut" lever that closes down one and sometimes two halves of a split nut to grasp the leadscrew and provide a drive for screwcutting.
Apron design can be roughly divided into "single-wall" and "double-wall" types. The "single-wall" apron has just one thickness of metal and, protruding from it (and unsupported on their outer ends) are studs that carry gears. The "double-wall" apron is a much more robust structure, rather like a narrow, open-topped box with the gear-carrying studs fitted between the two walls - and hence rigidly supported at both ends. This type of construction produces a very stiff structure - and one that is far less likely to deflect under heavy-duty work; another advantage is that the closed base of the "box" can be used to house an oil reservoir the lubricant ion which is either splashed around or, preferably, pumped to supply the spindles, gears and even, on some lathes, the sliding surfaces of the bed and cross slide as well.

COMPOUND SLIDE REST consisting of the CROSS SLIDE and TOP SLIDE
Sitting on top of the "Saddle" is the "Cross Slide" - that, as its name implies, moves across the bed - and on top of that there is often a "Top Slide" or "Tool Slide" that is invariably arranged so that it can be swivelled and locked into a new position.
Very early lathes had a simple T-shaped piece of metal against which the turner "rested" his tool (all turning being done by hand) but when it became possible to move this "Rest" across the bed by a screw feed it became known, appropriately enough, as a "Slide-rest". The earliest known example of a "Slide-rest" is illustrated in Mittelalterliche Hausbuch, a German publication of about 1480.
After the "Top Slide" became a more common fitting the term "Slide-rest" was not so frequently used - and the different functions of the two slides led to their specific names being more widely adopted.
When two slides are provided (or sometimes, on watchmaker's lathes, three) the complete assembly is known as the "Compound" or "Compound Slide" or even "Compound Slide-rest". Some makers have been known to label the "Top Slide" as the "Compound Rest" or even the "Compound Slide" - but as "to compound" means the 'joining of two or more' - not 'one' - this use of the term in incorrect. The top and cross slide together should be referred to as "the compound"

CARRIAGE
The whole assembly of  Saddle, Apron, Top and Cross Slide is known as the "Carriage". Some American publications (even makers' handbooks) have been known to casually refer to this as the "Saddle" - but this incorrect.

HEADSTOCK.
The lathe Headstock used, at one time, to be called the "Fixed Headstock" or "Fixed Head", and the rotating shaft within it the "Mandrel". Today the mandrel is usually called the "Spindle", but this can cause confusion with the tailstock, where the sliding bar is known variously as the "ram", "barrel" - and "spindle".
The headstock is normally mounted rigidly to the bed (exceptions exist in some production, CNC, automatic and "Swiss-auto" types) and holds all the mechanisms, including various kinds and combinations of pulleys or gears, so that the spindle can be made to turn at different speeds.

HEADSTOCK SPINDLE
The end of the headstock spindle is usually machined so that it can carry a faceplate, chuck, drive-plate, internal or external collets - or even special attachments designed for particular jobs. In turn, these attachments hold the workpiece that is going to be machined.
The "fitting" formed on the end of the spindle is normally one of five types:
      1) - a simple flange through which threaded studs on a faceplate or chuck (for example) can pass and be tightened into place with nuts. This is a secure method, and allows high-speed reverse, but is very inconvenient on a general-purpose lathe.
      2) - A threaded nose onto which fittings screw. This is perfectly acceptable for smaller lathes, but unsatisfactory on larger industrial machines where, for reasons of production economy, the spindle may need to be reversed at high speed. Reversing a screwed-on chuck causes it to unscrew - with potentially disastrous results.
      3) - A "D1-taper Camlock" fitting - a long-used, standard system that employs three or more "studs" that are turned to lock into the back of chucks and faceplates, etc.
      4) - A  taper - either of the simple Hardinge type or, for bigger lathes, the "taper-nose, long-key drive" - an older but excellent American design where a large screwed ring was held captive on the end of the spindle and used to draw the chuck, or other fitting, onto a long, keyed taper formed on the spindle end. An ideal system for the rigid mounting of heavier chucks, it has now largely fallen into disuse. The fitting was available in various sizes starting at L00 (L zero zero) and worked up through L0, L1, L2, etc.
      5) - various fittings that became increasingly complex and apparently invented for the sake of being able to claim a National Standard (the famous not-invented-here syndrome). All these succeeded in doing was to raise manufacturing costs by preventing the interchange of spindle-nose tooling between machines and requiring firms to keep larger inventories of spares and numbers of duplicated fittings. Some of these efforts included: British and ISO Standard Spindle Noses - Direct Mounting; British & ISO Short Taper with Bolt or Stud Fixing; British & ISO Short Taper with Camlock Fixing; British & ISO Short Taper with Bayonet Ring Fixing and, of course, German Standard Spindle Noses. Unbelievably, there appears never to have been a French standard - and we still await an official announcement of the rumoured Botswana-Standard Triple-cam with Over-locking Nose and Chinese-designed New Moon Slide-and-Snap-Approximately fittings.

BACKGEAR
As its name implies, "backgear" is a gear mounted at the back of the headstock (although in practice it is often located in other positions) that allows the chuck to rotate slowly with greatly-increased torque (turning power). Backgeared lathes are sometimes referred to a "BG" or "BGSC" - the latter meaning "backgeared and screwcutting". At first, the ability to run a workpiece slowly might seem unnecessary, but a large-diameter casting, fastened to the faceplate and run at 200 rpm (about the slowest speed normally available on a lathe without backgear) would have a linear speed at its outer edge beyond the turning capacity of a small lathe. By engaging backgear, and so reducing the speed but increasing the torque, even the largest faceplate-mounted jobs can be turned successfully.
Screwcutting also requires slow speeds, typically between 25 and 50 rpm - especially if the operator is a beginner, or the job tricky. A bottom speed in excess of those figures (as usually found on most Far Eastern and European machines but not those built in the United Kingdom) means that screwcutting - especially internally, into blind holes - is, in effect, impossible. These lathes are advertised as "screwcutting" but what that means in reality is just power feed along the bed. Even if you go to the trouble of making up a pulley system to reduce the spindle speeds you will find the torque needed to turn large diameters at low speeds causes the belts to slip. The only solution is a gear-driven low speed and so a proper small lathe, with a backgear fitted, not only becomes capable of cutting threads but can also tackle heavy-duty drilling, big-hole boring and large-diameter facing: in other words, it is possible to use it to the very limits of its capacity and strength.
Beginners are sometimes confused about how to engage backgear - especially if the lathe lacks a handbook - but with a little care anyone can work out how it should be done, at least on a conventional machine. On the main spindle of the lathe, the one carrying the drive pulley, will be found a large gear, generally referred to as the "Bull Wheel". The Bull Wheel is attached to the pulley by a nut and bolt, a spring-loaded pin, a pawl that presses into a gear on the pulley (or some other means) and, if this fastening is undone - by slackening the nut and pushing it towards the pulley, or by pulling the pin out - it should be found that the pulley will spin freely on the shaft. By moving the "backgears" into position - they generally slide sideways, or are mounted on an eccentric pin - the mechanism will come into operation. If the pulley will not spin on the shaft, or there seems to be no obvious way of disconnecting the Bull Wheel from the pulley, it may be that you are dealing with an "over-engineered" machine where some clever device has been introduced to make life "easy" for the operator. Sometimes there will be a screw, flush with the surface of the drive pulley and beneath this a spring-loaded pin that pushes into the back face of the Bull Wheel. Quick-action "Sliding-cam" mechanisms are occasionally used (as on the Drummond and Myford M Series lathes) where a knob on the face of the Bull Wheel has to be pushed sideways, and so ride up a ramp, which action disengages the connecting pin automatically. Some lathes, with enclosed headstocks (like later Boxford models) have a "single-lever" backgear; in this system moving the first part of the lever's movement disengages the connection whilst the next brings the backgear into mesh.

LEADSCREW
Originally termed a "master thread", or described as the "leading screw", but now always referred to as the "leadscrew", this is a long threaded rod normally found running along the front of the bed or, on some early examples running between the bed ways down the bed's centre line. By using a train of gears to connect the lathe spindle to the leadscrew  - and the leadscrew to the lathe carriage - the latter, together with its cutting tool, could be forced to move a set distance for every revolution of the spindle.

TAILSTOCK
The Tailstock was once known in England as the "loose stock",  " Ppoppet head" or "loose head" - the latter old-fashioned term being used by Harrison and other English firms in some of their advertising literature until the early 1970s. The unit is arranged to slide along the bed and can be locked to it at any convenient point; the upper portion of the unit is fitted with what is variously called a "barrel", "spindle" "ram" or "shoot" that can be moved in and out of the main casting by hand, lever or screw feed and carries a "Dead Centre" that supports the other end of work held (by various means) in the headstock.
Special centres, which rotate with the work, can be used in the tailstock ; these are known as "Rotating Centres" and should not be referred to as "live centres" - that term being reserved for the centre carried in the headstock spindle.
Long ago centres were referred to by turners as "Poppets" - presumably from "pop it in" - and they carried their own with them, secured in cotton waste and jealously guarded in the top pocket of their overalls.

COUNTERSHAFT (in the USA sometimes referred to as a "jackshaft")
Most small electric motors in Britain run on a 50Hz supply spin at 1425 rpm, while those in the USA (on 60Hz) are usually marked rather faster at around 1700 rpm or so.
If the lathe spindle was to be driven directly from one of these motors, even using a small pulley on the motor shaft, and a larger one on the lathe, it would be turning far too quickly to be useful for the great majority of jobs; hence, it is necessary to introduce some way of reducing the lathe's spindle speed  - and that is the job of the countershaft.
In a typical arrangement, illustrated here, the motor is fastened to an upright, hinged, cast-iron plate and fitted with a small pulley on its spindle. Because the 1500 rpm motor is driving a much larger pulley in a ratio of something like 5 : 1 - the speed is reduced to 300 rpm (1500 divided by 5).
On the same shaft as the very large pulley is a set of three smaller pulleys, arranged in the "reverse" order from those on the lathe. If the middle pulley on the countershaft is made to drive the identically-sized pulley on the lathe spindle that too, of course, will turn at 300 rpm. The pulleys each side of it are normally arranged to halve and double that speed - hence the creation of a speed set covering a useful 150 rpm, 300 rpm and 600 rpm.
It is a simple matter to fit both a small and a large pulleys to the motor shaft, and two correspondingly larger pulleys on the countershaft, and so double the number of available speeds to six. If a two-speed electric motor is used the range doubles again to 12 and, should the lathe designer have managed to squeeze a four-step pulley between the spindle bearings, a total of 16 would be available; with a backgear fitted the total would rise to thirty-two speeds that, typically, might start at 25 r.p.m. and extend all the way up to over 3000 rpm.

CHANGEWHEELS and TUMBLE REVERSE
Changewheels are the gears used for screwcutting and power feeds; they take the drive from the headstock spindle down to the leadscrew. They are normally contained within a cover at the extreme left-hand side of the lathe - but many older lathes, built in times when manufacturers were not concerned with saving people from their own carelessness and a loss of fingers, left them exposed.
Called "changewheels" because engineers once referred to "gears" as "wheels" and the word "change" because of the necessity to change them every time a different thread, or rate of tool feed, was required, the expression goes back to the earliest time that gears were used for this purpose.  The set of gears - called a gear train - is usually carried on a quadrant arm that can be adjusted by being swung on its mounting to allow the mesh of the topmost gear mounted on it with the output gear (called the stud gear) on the spindle (or tumble reverse mechanism) to be set. In the UK the arm is sometimes referred to as the "Banjo" - although this expression should really be limited to those types with just one slot. Some manufacturers, to make life difficult for themselves and their customers, tried other systems as well. A drive through changewheels often incorporates a tumble-reverse mechanism by which means the drive to the leadscrew can be instantly reversed - and hence the cutting tool made to move towards or away from the headstock at will. In its "neutral" position it also allows the headstock spindle to rotate freely and quietly without having to drive the screwcutting changewheels and leadscrew.
For more details of screwcutting, click here and for a further explanation of the features required on a small here..