Whitewater boats are usually made of either fiberglass and/or kevlar, or more commonly, rotomolded plastics. The later are not as fast or manueverable as the lighter, more agile cloth/resin boats, but they are practically indestructible and a lot of fun. An ultra-low-volume kayak is sometimes called a squirt boat; a very short kayak with blunt ends is sometimes called a bat boat (it looks like a suppository with a cockpit). K-1s are the rule in whitewater, and you rarely see K-2 slalom boats.
Speaking of slalom boats, they're halfway between ordinary river-running craft and squirt boats. They're 4 meters long (because that's the minimum allowed length), and have very flat decks (to sneak the ends under the poles). They are built to optimize speed and agility at all costs...including stability.
A relatively recent innovation in kayak construction is the "funyak" or "ducky"; these are essentially one-person self-bailing rafts in the shape of a kayak. What they lack in maneuverability they make up for in stability; they're an ideal craft for a beginner interesting in solo paddling, as they allow folks to get a taste of whitewater without developing skills such as the eskimo roll, eddy turn, etc. They're also used by experienced river runners as well, and can be paddled anywhere a decked kayak can. However, since they can't be eskimo-rolled, they may not be appropriate for some big-water situations.
Some generalizations: Fiberglass kayaks tend to be lighter, faster, and more costlier (~$1500). Rotomolded kayaks are heavier, slower, yet cheaper and virtually indestrucible (~$750). Portable kayaks are heavier, wider, and very expensive (~$2000) but great for travel on planes or if you live in an apartment. Not as low maintenance as fiberglass though. PVC inflatable kayaks are light, cheap (~$500), and easy to transport. Not much storage space and prone to punctures, but easy to fix. Although designed for warm water, they have been successfully paddled on long trips in Alaska. Wooden kayaks are usually kits, and are fairly light and durable but require more maintenance (~$600 for a kit).
Before you pick a kayak, decide what you'll be doing with it. Camping, fishing, photography, day trips, aerobic workouts, expeditions, racing, surfing, etc. Also consider your size in relation to the boat's size. You should comfortably fit in the boat, not too snug and too loose.