There are many different boards all for different reasons. Shorter surfboards (directionals), twin tips, doors/wings, race boards, skimboards, landboards, snowboards, etc... What board to get? Well similar to kites, having a quiver of boards is also what ends up happening.
When first getting into kiting and trying to decide what board to look for is confusing as well. Boards range in price ($300 to $1k+) and if on a budget, can get them used as well. Rule of thumb learning to kite is lite winds, big kite, and big board. Big boards allows more float (stay up on plane longer) when trying to figure out how to properly control the kite.
Recomendations (depending on your weight) is start with a big (lite wind) board, like that of a wing or a door. Which is in 140cm to 160cm or larger range. When you get this board and get past your water starts and get riding figured out, then you can keep this large board for your 'lite wind' days, and add your 2nd board. Reason why you would want to get a large board to start out with is couple reasons.
1st would be early plane up on the water and stay up on plane. With our gusty on/off/on winds you will work the kite more figuring things out. If you try and start learning on a small board you will be sinking into the water everytime you blow thru a power stroke on your kite, or when wind is not strong enough for a brief time (what we call a hole or inconsistant wind). You will be fighting the kite to stay up on plane. Then the opposite could/will happen, you will power stroke the kite when a gust hits and could get yanked hard up over the board to a face plant, and dragged for brief distance.
2nd there is alot going on when first getting your water starts figured out, with a board that can easily stay up ontop of the water, you can primarily work on kite skills. After many hours/days/weeks/months depends on how frequently get out and kite you can 'then' progress to a smaller board.
3rd reason is lite wind boards are designed to help you stay upwind easier. Once you go down to a smaller board you will need to be in more of a powered up kite condition, and work more on board edge control. Unless you like to do alot of walking back upto the kite launch, starting out with a bigger board is a wise investment and allows you more Time On the Water (TOW).
Normally your 2nd board to get will become your 'main board' after you get thru your 5stages of progression, which is a smaller twin tip board (trick/freeride/all around board). This board looks similar to a wakeboard, but is 'not' a wakeboard. Twin tip kiteboards are flatter boards typically with concave or double concave. Twin tip sizes (again depending on your weight) ranges from 120cm to 140cm.
Other boards include surfboards (directionals) to do just that, surf. Yes we surf in Kansas at cheney lake during gale force winds on tiny kites that the wind kicks up 5+foot rollers.
Race boards are directionals similar to surfboards yet are fatter designs with square tails and long fins. These are the boards that are used in course racing.
Skimboards are strapless flat boards (no fins) to ride right up along a shoreline in very shallow water right upto ankle or 1" deep. Performing fun skate style tricks like: ollies, shove-its, kick-flips, etc... similar to skateboarding.
Landboard is a large mountain board with feet strap hoops, very wide and strong trucks and inflatable tires to ride in open fields. We typically ride landboards on soccer fields and in the fall/winter when cold season or lake is iced over but no snow to snowkite on.
Snowboards is just that. We kite with them occasionally in the winter time.