KiteJan

Diamond

What most people 'of a certain age' think of as a kite - these are still diamond shape and, yes, do have a tail. But with the 2 lines they can be flown around the sky in a standard 2 line way. The cheapest ones are just made out of plastic (and believe me don't last long), the better ones are in a nylon fabric and will last longer. 

Cheap, fun, children can fly then and they look nice in the sky (the kite, not the children) - just can be hard to launch by yourself and don't expect to do many radical tricks.....

Delta

What is now thought of as the standard shape for kites - they come in a huge range of sizes and variations of the standard Delta shape. Some are better suited at steady flying - being able to fly in precise formations and are very controllable. Others are better at radical flying, where they are actually slightly unstable in the sky and can do all manner of exciting tricks. Easy enough to launch by yourself with practice, but the spars can (and do) break if crashed a lot (all can be repaired, by the way). Just about all of them are made out of Nylon fabric (such as ripstop, or better) with either fibreglass (on the cheaper ones) or carbon fibre (on the more expensive ones) spars.

Parafoil

These are soft kites (i.e. with no sharp, breakable bits). They can fold up small, work in all kind of winds from light to strong, and can do lots of manoeuvres. However they can't do radical tricks (unless you count turning inside out as a trick of course), nor are they really formation flying kites (Deltas are generally better). The only difference between a 2 line Parafoil Stunt kite and a 2 line Parafoil Power kite is really a) the size of the kite and b) the strength of the wind you are flying in! It seems that any parafoil kite over 2 square meters is going to be classed as a power kite really.