A crochet hook is, basically, a stick with a "crook" on the end of it, used to work crochet. Simple.

There are four parts to a modern crochet hook: The point, throat, shank, and thumb rest.
        ___          Thumb rest
point  < <____,,,,,___/_,,,,,,,,,,,,,,
        \_________,,,,,,______________)
          shank
And the throat is the "<___" area, where the inside of the hook meets the shank.
The point is the tip of the hook, where it is inserted into previously made stitches to create new stitches. A fine balance of bluntness and sharpness must be achieved with the point. A point that is too sharp can separate the threads of yarn, making the work messy, and a point that too blunt can be very difficult to work into a stitch.

The throat of the hook is important, in that it allows you to slip loops off the hook while maintaining other loops on the hook, the primary principle of crochet. The cut out area of the throat must be large enough to hold yarn of the size you are using without overflow, while it must also be small enough that it will not swallow all of the loops on the hook, preventing you from taking any off.

The shank is the simplest part of the hook. It is the long "stick" where loops sit until you manipulate them. The most important feature of the shank is its diameter: the size of your shank will determine the approximate size of the stitches made with that hook.

The thumb rest is a flattened area on the hook which serves two purposes: Firstly, it allows you to rest your thumb against the hook with less risk of it slipping off the rounded surface. The other use for the thumb rest is that it serves as a stopper for where loops can sit on the hook. Usually, no loop will sit on the hook past the thumb rest, so no loop will slip off the end.

Hook sizing is annoying and confusing. Yeah. But necessary. Steel crochet hooks are sized like this, from biggest to smallest:

U.S.               1  | 2  | 3  | 4 | 5 | 6   | 7  | 8   | 9 | 10  | 11 | 12 | 13 | 14
English           3/0 |2/0 |1/0 | 1 |1.5| 2   |2.5 | 3   | 4 | 5   |5.5 | 6  |6.5 | 7
Continental (mm)   3  |2.5 |    | 2 |   |1.75 |1.5 |1.25 | 1 |0.75 |    |0.6 |    |
Hook sizing for plastic and aluminum hooks go like this, from smallest to largest:
U.S.               1/B  | 2/C | 3/D | 4/E | 5/F | 6/G | 8/H| 9/I | 10/J| 10.5/K
English            12   | 11  | 10  |  9  |  8  |  7  |  6 |  5  |  4  |  2
Continental (mm)   2.25 |2.75 |3.25 | 3.5 |3.75 |4.25 |  5 | 5.5 |  6  | 6.5
A final note: lots of patterns nowadays call for specific hook sizes, and its enough to make your head spin. Truth be told, use the hook that is comfortable for you and keeps gauge. That's all you need.