In almost all cases you shouldn't ride on a trail when it is muddy. When you ride in the mud you are significantly contributing to trail erosion, and you seriously effect the quality of the surface of the trail when it dries out.
Trail damage and erosion is one of the most effective reasons people use to get trails closed. You shouldn't be surprised to find a trail that has been ridden in the mud one day closed to mountain bikers another.
A mountain biker should always make a conscious effort to preserve the environment they are riding in. This includes an honest evaluation of the trail condition and an effort to minimize damage to it.
That said, there are some other issues to riding wet trails.
Damp trails are particularly fragile, so, as always, avoid hard braking and locking up your wheels.
Don't go around puddles, go straight through them. If everyone goes around the outskirts of a puddle it damages the sides of the trail and widens it. Going through the puddle keeps the trail at the same width and minimizes trail damage.