In England they're called "chainsets." In the U.S., "crank" and "crankset" usually refers to the crank arms and assembled chain rings.
Most decent cranks are made of aluminum, although some are steel, some are titanium, and now a bunch are carbon fiber.
Aluminum cranks can be machined (carved), forged (smashed), or cast (poured into a mold).
Cranks are usually differentiated two ways: 1. By bolt circle diameter (bcd), and 2. By length.
1. Bolt-circle diameter.
Chainrings attach to right-side cranks by chainring bolts, so the chainring bolt holes in the chainring have to match up to the corresponding holes in the crank. If you draw a circle through the centers of all the holes and measure the diameter of that circle, that's the bolt circle diameter (bcd). The smaller the BCD, the smaller the chainring that'll fit on it. Here is a list of common BCDs from the past and present, and their respective minimum-sized chainrings:
Most road cranks designed for two chainrings have BCDs of 130mm. A 50-tooth chainring designed for such a crank is designated thusly: 130 x 50t. Etc.
Campagnolo is the odd duck here, with BCD of 135mm.
Good manners suggests that manufacturers make their cranks (of a type, say road doubles, or whatever) to the same standard as everybody else. That way, you the rider can change chainrings and have a great selection (from many makers) to choose from. But manufacturers sometimes have bad manners, and make up their own, unique BCD so if you buy their crank, you'll also have to buy their chainrings, which are generally more expensive. Eventually a specialty chainring maker will see a market and make a lower-priced odd-BCD chainring, but it's still a mean thing to do in the first place. Standards do change over the years, and a manufacturer can build to its unique standard by saying the others should switch, or that theirs is better, or whatever. We don't buy it.
Road cranks designed for three chainrings (called "road triples") have a secondary BCD to accommodate a smaller ring for an easier gear. This type of crank will have a BCD specification something like: 130 x 74, indicating it fits outer and middle chainrings with a 130mm BCD, and an inner chainring with a 74mm bcd. A common chainring combination for such a crank is 50t (outer) x 40t (middle) x 30t (inner); although that is one of many possible combinations.
Mountain bikes cranks have BCD of either 110 x 74, or 94 x 58.
There is absolutely no reason why you can't use a mountain bike crank on a road bike, and one advantage in doing so is that you can use a middle ring smaller than 38t (which is the smallest you can use with a normal 130mm bcd road crank). So, you can set up your bike with a 50 x 34 double chainrings, or 46 x 36 x 24 triple chainrings, or many other combinations not possible with the road-standard 130mm bcd. But as a rule, cranks specifically designed for mountain bike use tend to have higher Q-factors (wider between the pedals) than road-specific cranks. There are exceptions, though.
2. By their length.
In the old days, you could find cranks between 150mm and 185mm in length, often in 2.5cm increments. Even as recently as the early 80s, any crank maker was expected to offer 165, 170, 172.5, and 175. But since then, and perhaps influenced by the trend away from a road-dominated market to a mountain-bike dominant one, cranks outside the 170 to 180 range are hard to impossible to find, and many makers offer only 170s and 175s.
Conventional wisdom says shorter cranks are easier to turn quickly (spin), and longer cranks offer more leverage, which is better for climbing hills and pushing big gears. And, that short-legged riders should pedal shorter cranks.
All we will add to that is the warning that riders of average height and leg length will adapt quickly enough to any length between 170 and 175; and that even experienced riders would be nervous if their life depended on being able to tell a 2.5mm difference in crank length; and that knee experts generally agree that shorter cranks, in some cases, may be easier on your knees.
Q-Factor. Yes, Q-Factor.
It's the distance between the pedals at the outside of the cranks. In other words, it's how far apart the cranks put your feet. Are you walking a tightrope, or giving birth? Pedaling a bicycle or walking like a duck? That's Q (quack)-Factor.
At the turn of the century, this crank dimension was called "tread," a term still adhered to by some. We call it Q-Factor because when "tread" was in vogue, tires were all smooth, and "tread" couldn't possibly have referred to the rubber pattern. But these days, "tread" means rubber pattern, so the discussion can't begin until you dig out of that hole.
In any case, Q varies from about 135 to 180, with most good road double cranks hovering around 140mm, and most mountain bike cranks around 173mm.
With the exception of Ritchey, crank makers do not list Q. Most make high-Q cranks, and would prefer that you not ask. There can be no arguing that a lower Q improves aerodynamics, but a far more compelling and practical reason for attending to your crank's Q is...how it feels.
Some riders are not sensitive to it at all. Others have physiological quirks that make them better off with their feet closer together (low-Q) or farther apart (high-Q) cranks. In general, road riders accustomed to the superb, low-Q professional grade Campy, Zeus, TA, and Stronglight cranks have a harder time adapting to modern cranks with higher Qs. They often hate them, and don't even realize what it is until somebody suggests the Q. On the other hand, riders raised on higher Q cranks often think the whole issue is silly, because they can't tell any difference.
If you're used to a low Q crank and you pedal a bike with a high Q, it's possible that you'll experience pain on the inside of your knees.