Skeet & Trap are both shooting sports that involve shooting clay targets. The difference is how the games are shot. Unless your talking about international Trap or doubles, you only need one round in the gun for trap.
Skeet requires a gun that can fire two shots in quick succession. Skeet can be played with a pump gun (I've seen people that can do it well) but it takes a lot of skill.
A lot of people (incorrectly) refer to any game involving flying clay targets as "Skeet".
As for the 870 / 887 / Supernova choices, those are all good guns with different strengths. Unless you're going to do nothing but stand in the rain and shoot waterfowl; I'd recommend the Remington 870.
The 870 is far more an "all around gun" than the 887 or Supernova.
The 887 and Supernova are designed for low maintenance in wet conditions and have 3 1/2" chambers that are favored for users of steel shot.
Steel shot is mandatory for waterfowl hunting but steel pellets weigh less than the SAME size lead pellets. So to compensate for the reduced mass of steel shot, one uses larger pellets and that requires MORE pellets in the shell to get the same pattern density. That, in turn, requires a longer shell to hold the bigger pellets. 3 1/2" chambers are good when using steel shot but are not needed outside of waterfowl hunting. The 3" chamber on an 870 magnum is a good compromise and of course it can shoot 2 3/4" shells as well.
The 870 has been in continuous production since 1950 and millions have been made. The 870 has been used by the military, law enforcement, hunters and sport shooters for decades. IT WORKS!