Huge generalisations follow. Handle with care.

I guess you can roughly sort cymbals overall into two categories, and each of those into a further two:

1. Timekeeping. Hi-hats and rides.

Generally used to mark time, but in different ways.

Hi-hats have a similar construction to crashes, but it's the combination of that plus the way the two cymbals interact that determine the sound. Most commonly with a heavier-weight cymbal on the bottom and a medium on the top, the way the cymbals interact determine the sound that's made when you strike them when held together with the hi-hat pedal (with varying degrees of opening), the "chick" sound when you close them sharply, and the "splash" sound when you close and then allow them to open quickly. They're capable of a huge range of colours and textures.

Ride cymbals are generally large, with the commonest sizes being 20, 21 and 22" but can be found as large as 30" and there's an overlap category of "crash/ride" cymbals which I guess cover roughly 18, 19 and 20" cymbals. Shaped like large crashes, they tend to be heavier (some very heavy indeed). They can be generally said to have two important areas for sound - the bell and the bow. The bell produces the defined "ting" sound that cuts through the music. Riding the bow has a less pronounced stick sound, but backs it up with a wash of sound that "cushions" the music. The construction technique of the cymbal (metal composition, weight, bell size, lathing, hammering and so on) determine the exact effect achieved - perhaps higher-pitched with more stick sound and less wash for rock music, or a less pronounced stick sound, darker complex sound and longer wash for jazzier music. Crash/rides tend to be thinner and as their name suggests they can be crashed as well as ridden.

2. Accents. Crashes and effects cymbals.

Used to "punctuate" the music somehow, providing accents and other colours.

Crashes generally provide a longer, sustained, arguably more "musical" sound. Coming in various sizes from 14" through 20" typically, the amount of metal and treatment techniques (hammering, lathing, polishing) influence the sound you get.

Effects cymbals cover a multitude of bases, but possibly the commonest two categories are splashes and chinas. Splashes tend to cover the size range from 6" through 12" and are generally constructed similarly to regular crashes but tend towards thinner cymbals. They have a short-lived, glassy sound that provide a nice contrast to the fuller crashes. Chinas are "trashier" cymbals with their treatment and the "folded back" edge making the sound explosively shorter, darker and more complex. Just to show you that it's dangerous to generalise, chinas also make useful ride cymbals - Simon Phillips, for example, uses his mostly for riding, not crashing. There's also an overlap with splashes, with the cymbals known as "china splashes" which combine the two characteristics, giving generally a very short, complex punch of sound.

There are others, of course, from various bell cymbals through "stacks" (two cymbals placed on top of one another) and specialty cymbals like Sabian's "O-zone" crash with six immense holes in it (a remarkable cymbal indeed). And as I say those are generalisations. You can ride crashes (common in rock music), strike the bells of any cymbal for an accent, mount your hi-hats upside down ...

Go to a store, and play a few.