Collets are used for holding either tools or small work.
Let's say you wanted to make a small screw from 1/8" screw stock. Putting that into a 4-jaw chuck, dialing it in and so on gets incredibly fiddly and wastes quite a bit of time - if you're able to do it. Many larger chucks can't grab hold of very small diameter parts.
Enter collets. Let's start with the 5C collet and the South Bend 10L or similar 5C lathe (which would be 1 3/8ths through the headstock) and you want to turn some screws, pins or other small parts.
You'll pull off the 4 or 3 jaw chuck - either unscrew it or un-cam it.
This leaves you with a "naked" spindle.
You'll have a collet adaptor that slips into the spindle at the working end. You'll have a tube, into which the 5C collet can screw, that goes into the headstock from the left side. It will have a collar on it to allow the hollow shaft to spin inside the collar that trues up the tube where it goes into the headstock. They sort of look like this:
While that's for a Monarch 10EE (which is a very good toolroom lathe), there are closers which are the same form for South Bend 10L's and 13's.
With some talent and time, you could make your own screw-type collet closer. Been there, done that. Look in Machinery's Handbook for the thread and size specs. You want to use 4140 alloy steel tube.
OK, so you have the tube through the headstock. You put the collet into the adaptor that you've put into the headstock. Now you put your part into the collet of the correct size (for 5C collets, this means you choose a collet that's no more than +/- 0.005 to +/- 0.008 or so away from your part size). You put your collet (with the part in it) into the adaptor in the headstock, then you start tightening the handwheel on your left until the collet engages the threads in the tube and the collet gets sucked into the adaptor in the headstock, and the collet crushes down on your part.
The beauty of collets is that if you have a good set of collets, a true spindle and adaptor, you don't have to dial anything in. The part in the collet will be within 0.001 of being dialed in. THAT is the real beauty of collets: You can put parts into them and not spend a second dialing the part into true.
There is a class of lathe that worked especially well with collets - so much so that the lathe was optimized for using collets. The Hardinge HLV-H is one such example of a lathe:
The closer on the left side of that lathe could be adjusted so that you just shoved the lever to the left, the collet locked onto the part and away you went. When you were done, you'd grab the closer lever, jerk it to the right, and the part came free.
3C and 5C collets were designed to hold work, but could also hold tools. One of the things many gunsmith wanna-be's don't know is that there used to be gunsmiths that didn't own a milling machine at all. They had a milling attachment that mounted onto the compound or cross-slide of the lathe. It was basically a machinist's vise with an adjustment screw that would give you a Y-axis adjustment with a calibrated collar on the screw, you'd use your cross-slide for the X and your left/right movement of your saddle (or the compound if you turned it to parallel with the spindle) for your Z axis.
You'd put a collet into the headstock, then mount your end mill in the collet, wind it up and start milling.
5C collets are threaded on the outside, and were intended to have tube with internal threads grab the collet, so you could put long, thin workpieces into the headstock. Screwstock rods were quite commonly turned into screws like this on a lathe.
Now, R-8 collets were made for holding tools - and became "standard" for Bridgeport-type milling machines after WWII until recently. The R-8 collet has an internal thread for the drawbar on a mill to pull the collet into the spindle, so you can't put long work into the collet. It was designed to accept straight-shank end mills and such.
ER-type collets (designed in Germany since CNC machines became very common) are usually used for holding tooling in CNC machines. But they clamp onto tooling better than R-8 collets do (you can slip mills and drills in R-8 collets under heavy loads) and they clamp over a wider range of sizes than R-8's or 5C's. ER-type collets hold tools in CNC spindles under much, much more horsepower than any Bridgeport or small engine lathe can hope to develop. I've put tools into ER-40 collets in a VMC then loaded up the spindle to about 10+HP - indicated on the load meter, and the tool never slipped. R-8 collets I've had slip when I've used large diameter tooling in a Bridgeport-type mill.
A good set of 5C collets in a very close set of sizes (eg, by 32'nds of an inch) can be quite spendy. 5C collets are also available in square bores, hex bores, and so on. They're very, very useful... but for most people, a set of good 5C collets and all the associated tooling can be rather expensive - just a complete set of round hole 5C's can run over $1K from Lynden (which makes some of the best collets out there), whereas you can get into a good set of ER32 collets for a lot less money because they're much more common in modern CNC shops. If you can find used 3C/5C/16C/R8/etc collets that were made in the US, you can find bargains... but sometimes, collets get abused by their owners and used collets can be sprung, cracked or mangled on the face of collet.
How important are collets in a gunsmithing operation? Well, let's put it this way: You'd end up making something very similar to collets if you don't have them. There's just too many small parts and tools you need to grab hold of in gunsmithing. In a big machine shop, where you're working on workpieces so big you need a jib crane just to get your chuck on/off the lathe, you'll almost never see collets in operation.
5C collets are also used in "collet blocks" to hold parts in mill vises. There are square and hex collet blocks, so if you need to machine something four or six times, you just set up a mill stop on the vise, then put your workpiece (in the collet, which is held in the collet block) into the vise, you do your operation, loosen the vise, turn the block, repeat, etc.
Here's an example of a collet block for 5C's: