Trap: You stand 16 yards behind the trap house. The launcher oscillates left & right so you don't know the angle the bird will be launched. You have to quickly track the bird and determine the lead angle. Shot typically with #8 shot, sometimes with #7.5. The point where your shot intersects the bird is 34-35 yards away, so you typically use MOD, IMP MOD, or FULL chokes. Trap also has a Handicap phase startong 20 yards back of the trap with better shooters being even farther back (all the way to 27 yards back). Last phase of Trap is the Double phase where the launcher is fixed and throws two birds simultaneously. Usually shot with 12 gauge.
Skeet: Two launch houses, one high and one low. You start under one house and work your way around in semi-circle to the other house. Birds are primarily crossing birds, and engagement is typically no more than 25 yards away, sometimes closer. #9 shot and open chokes (Skeet chokes) are used. Regulation Skeet is shot in 12, 20, 28 gauge, and .410 bore.
Clays: Can be all of the above and more, typically in a woodland environment where you walk from station to station. (Golf with a shotgun.)
Actually there is a lot more to each of these sports that the thumbnail descriptions above. Your hand launcher does what is called "Hand Trap" and is generally not as difficult as regulation Trap. There are also Olympic versions of these sports with smaller, faster targets, lighter shot charges, and sometimes the opportunity to shoot a second shot if the first missed (in Trap).
