Chainsets
Mountain bikes typically have a triple chainset. That is, one with three chainrings (or 'chainwheels') on the right-hand crank to give a very wide range of gears, so that the bikes can be ridden over all sorts of terrain. The right-side crank has four or five arms (known as the 'spider') and the chainrings are bolted onto these arms. (Cheaper chainsets may not have detachable chainrings).
The dominant name in all mountain bike parts is the Japanese manufacturer, Shimano
For a mountain bike the chainring sizes are usually 46, 34 and 24 teeth, though some riders might chose to alter this by a couple more teeth if they are particularly strong, or a couple of teeth less for lower gearing if they ride in very hilly areas or carry heavy loads.
Usually made from light aluminium, the dominant name in all mountain bike parts is the Japanese manufacturer,
Shimano, whose range covers everything from entry level recreational products (Tourney, Altus, Acera and Allivio), through the enthusiasts' needs (Deore, Deore LX) up to hard core riders and pro gear (Deore XT and XTR).
Cranks are made in different lengths. There has been much debate relating crank length to leg length recently, but since crank only varies by about 15mm (165-180mm) this is not a major issue. Very tall or very strong riders might prefer the longer length and shorter riders may find the 165mm cranks easier to use, but in reality most riders can manage very comfortably with 170 or 175mm cranks - as this really relates to the gear size being used rather than the rider's size.
Bottom brackets
Bottom bracket
The whole chainset assembly is then bolted on to the bottom bracket axle, which rotates on bearings, inside the bottom bracket shell of the bike.
For years this was a simple, square-tapered axle, but recently Shimano introduced a splined axle to interface with their cranks. This has various advantages: the axle can be made larger, stiffer and lighter, and the fitting should be more secure. However, if replacing your bottom bracket you do have to check that your cranks will be compatible!
Nowadays, the bottom bracket is usually a cartridge unit. These are simple to fit and require no maintenance. For real enthusiasts, exotic materials such as titanium are available to save precious grams of weight.