Man, this will be a long, long answer. I'll keep it as short as possible.
There is a high house and a low house. The targets emerge in a sequence, at your call, following a predetermined path and predetermined speed, the same, each and every time. You (optionally) pre-mount your gun because you know exactly where it will be and where it will be going each and every time, with variations for wind. Shooter stations around an arch of (almost) a half circle. Shooter progresses from one station to the next shooting either two singles, one from each house (3-4-5-8), or one single from each house and a double (1-2-6-7). Total targets is 25. Common to hit 24-25.
There is a trap house 17-25 yards in front of an approximate 30 degree arch where there are 5 shooter stations. The targets emerge going away from you either in singles or in pairs, in random angles. Shoot 5 and go to the next station. Guns may be premounted is desired. 25 shots is a round. Common to hit 23-25.
Most simulates hunting. Each course is customized. Although you know where the targets will emerge and the path they will take, you are most likely in the woods for many shots and the window of shot opportunity is sometimes small. There may be shots that resemble rabbits running and hopping, quail jumping, springing teal, ducks landing, or grouse or pheasant flushing. Guns may NOT be premounted. Sporting clay courses can be anywhere from 25-100 rounds. Mostly 50. Common to make 80-85 of 100.
I'll add 5-stand:
There are 5 'stands' side by side where shooters stand. Targets will emerge from a minimum of 6 locations, crossing, springing, ground running, outgoing, or incoming. You know the sequence and path, but again the course is customized each day usually. 25-50 is a round and 23-25 is common for the better shooters. Never played 5-stand, but if I recall the gun may NOT be pre-mounted.
Please, if I erred anywhere someone please correct me. Or feel free to add to it. This is bare minimum information...
The remington.com web site has a good presentation on skeet and trap, in the "reading room" page of thier site.
In American skeet you can mount your gun to your shoulder and point it at an initial aim point and call for the target.
In ISSU skeet, sporting clays, and I think 5-stand, your gun must be off the shoulder. For each game there is a certain point. Like ISSU skeet there is a line on your vest at the top of your hip bone. The gun stock must be below that line. In sporting clays it only needs to be below the armpit.
If I could add just a little to skeet. Low house on the left, high house on the right. The birds fly across your path from (low house) left to right, and from (high house) right to left. There are 24 targets, you have one gimme you can take anywhere on the course. If you miss a bird at some station and want to make it up, you use the gimme. I use it for doubles at the last station. (in the center)
A side note, when shooting skeet and you are standing next to the high house, you can VERY easily be hit with flying chunks of clay pigeons. Ouch!!! LOL