What spindle interface do I need?
Cranksets are installed on bottom brackets, so it goes without saying that the bottom bracket and crankset must be compatible. In other words, the spindle interface must be the same for the bottom bracket and crankset.
Take a look at the following articles in order to see the differences between each spindle interface, and make sure the interface of your bottom bracket matches interface of the crankset. Note that some cranksets come bundled with bottom brackets, so you'll be fine as long as you've checked that the bottom bracket is compatible with your frame.
Chains for bikes with 8 or fewer sprockets are the same width, however, as sprockets are increased to 9, 10 or 11, the width of the chain decreases. This means the chainrings on a crankset are tweaked to perfectly mate with chains of specific width. Since it would be cumbersome to refer to chains using their true width in millimeters, we just refer to them in terms of 'speeds' (ie. a 9 speed chain).
Make sure to select a crankset designed for the number of sprockets on your cassette and use the corresponding chain.
Cranksets come in a multitude of different chainring combination. Some cranksets come with three chainrings (triple cranksets), two chainrings (double cranksets), a single chainring or even none. The choice as to what combination is best for you completely depends on you and is usually a function of the type of bike you ride, where you ride it, and your physical capabilities.
When changing from a double crankset to a triple crankset (or vice versa) make sure you have a front derailleur compatible with the number of chainrings on your crankset.
All cranksets have two crank arms (drive-side and non-drive-side) of the same length. Crank arm length affect the final gear ratio and the pedaling movement. Since all modern bikes already have plenty of gears to choose from, the length of crank arms are more of an adjustment to the pedaling motion than anything else.
Crank arm lengths are measured in millimeters from the bottom bracket spindle to the pedal spindle, center-to-center. The crank arm length is usually marked on the inner side of the crank arms, and are matched to the geometry size of the frame (ie. XS frames will sport shorter crank arms than XL frames).