Kitesurfing has some very different styles and disciplines and most frequently it is the board that defines that style.
There is a number of different board types so let's get down to the subject.
First and by far the most popular - the twin-tip board. Twin tip board is the easiest board to ride and it is the first board for a learning kitesurfer. It is bidirectional, so you don't have to switch feet when you make turns. It looks very similar to wakeboarding board but is still quite different (usually less bent and "sharper").
Twin tip boards come in many different shapes and sizes. Standard board lenght is somewhere between 130 and 140 cm, but there are also boards 160 cm and bigger (plus extra wide) that are used specially in light wind conditions.
Further more, standard length twin-tips also differ among themselves although they all look similar to the untrained eye. They are usualy sold under one of the three labels: freeride, freestyle or wakestyle. Boards labeled as freeride are usually the cheapest and are tipically the best choice for beginners.
For some people twin-tip is all you need, but there will also be those who want to add a new flavor to their kiting. They have a couple of options to choose from.
The wave board (or kite surfboard) looks similar to shortboard surfboard, and as the name says it is the best choice for riding waves. It is unidirectional and fins are placed only on the rear side of the board (or the tail). When you make turns you have to switch feet (tack or jibe). It has more volume than twin-tip which gives it extra buoyancy, more floaty feeling and enables easier wave riding. They can be ridden with or without footstraps.
Hydrofoil boards are becoming more and more popular in the last couple of years. In hydrofoils, there is a wing beneath the board that dives through the water and lifts the rider above surface as he picks up speed. They are famous for being able to ride in the lightest winds.
Race Boards are another option for those who want to go out in light winds.
There are also options like skimboard, alaia board or wakeskate.