Dave pretty well has it right about the attitudes and demeanors of the various groups, but here's a quick rundown on the basics of each game;
- trap is shot by groups of up to 5 people who stand 16 yards or more away from a low-built "trap house". Each shooter stands on a "pad" that's in a line pointing towards the centrre of the trap house, and when a target is launched, it will come out directly in a line away from one of those lines. (The angles are random, chosen by a "wobble" mechanism in the trap). Each shooter calls for a target and gets 1 shot at it, then the next shooter to their right gets their turn. After 5 rounds from one position, everyone switches one position to their right.
- skeet can be shot by more people than trap, but is usually no more than 4 or 5. A skeet field is laid out like half of a clock face, with a "high" house at 9:00 and a "low" house at 3:00; you have 7 shooting pads arranged along the outside of that clock face, with an 8th position in the middle of the field, between the traps. (The traps launch the targets out at an angle, so they don't hit the trap houses.) Each shooter gets a series of targets from each house (one high, one low, and one double), then it's the next shooter's turn. Doubles are only given at stations 1 and 2, and 6 and 7, and each shooter will repeat the shot at their first miss; this gives 25 rounds for a round of skeet.
- sporting clays is meant to more simulate hunting than either trap or skeet, even though both trap and skeet were originally designed to give hunters practice at making shots on game birds. In sporting clays, you get a variety of different-sized targets, travelling at different speeds, in different flight paths. In one station, you might have to shoot at a low target travelling overhead, then at a large "rabbit" running across a clearing in front of you. (Sporting clays is one of the few shotgun games where you're expected to shoot at a moving target on the ground; they simulate rabbits by using a large clay target with a thick rim that is launched sideways and bounces on its rim along a given path.) You might see a station simulating ducks settling onto a pond, or springing teal flying straight up before they fly off.
If you like shotgunning, you owe it to yourself to give each of these games a try.
HTH
Stacey C.