I shoot all three and may be able to give you a quick and dirty overview of them. I personally like to shoot all three and shoot in a trap and skeet league during the week, and shoot in a sporting clays fun shoot each month.
Trap is like long range shooting with a shotgun and more like hunting pheasant. You stand a minimum of 16 yards from where the target is launched. It will come out of a trap house sunk in the ground when you call for the target. The clay will come out traveling up and away. The machine oscillates so you don't know what angle it will come out. Normally the targets are broken somewhere around 25-35 yards from the minimum distance. (16 yard line) Tighter chokes are the norm, modified, improved modified, and full.
Skeet is like hunting dove, a lot of crossing targets. It is shot with two traps, one set high on the right and another one set low on the left. (Actually the high is on left and low on right. Graybeard) The targets are set and will travel at the same speed and distance each time. The shooter rotates in a semi-circle changing the angles the targets are presented. There is a lot more gun movement in skeet and the targets can get close. Open chokes are the norm, cylinder, skeet, and improved cylinder.
Sporting clays are like hunting everything all in the same day. The targets can come from any angle and at any speed. They can come as singles, report pairs, (where the second target is launched at the sound of the first shot) and true pairs. (Where the targets are launched at the same time.)
One round of either trap or skeet is 25 targets. One round of sporting clays is normally 100 targets.
