First, don't do it:
- Stay off protected trails when it's raining or just after a rain storm.
- Stay off the trails during the wet season. NEMBA is a good guide for this: For the Middlesex Fells, March is Trails Closure Month. This is just a minimum guide, some years the wet season is a bit later. Some years it's longer or shorter.
- Check trail conditions online for the areas in which you want to ride. More and more sectors track trail conditions online.
Avoid "braiding." Proper etiquette demands that you ride stright through the puddle, not around it.
However, sometimes there's mud, even in the driest summer.
Dealing with mud is simple -- and counter-intuitive.
The best practice on a muddy trail is to walk through the mud. However, that may not be very reasonable -- if the mud is deep. The next best way to traverse a wet section is to ride directly through the mud. The purpose is to keep the muddy spot as narrow as possible. When riders and walkers go around the muddy section, they widen the mud and create more damage to the trail. This damage is sometimes called "braiding." As riders begin to widen a trail, it starts to look a bit like a braid.
Trail stewards want trails to remain on the trail. As soon as you start going off-trail you damage flora and fauna. One of the most fundamental rules of trail riding is to stay on the trail -- muddy or not. Never go off-trail.
Why? It's easier to fix, or heal, a deep rut rather than a wide swath of damage.
This is most important on trails that are well kept, in parks, etc. It's less critical on primitive trails, doubletrack, and fire roads. If you're on a fire road, and you come upon a muddy section, riding around it is fine; the road is already wide, motorized vehicles are already doing way more damage than a bike ever will. Riding around mud spots in non-protected areas is fine. Regardless, when in doubt, ride through or walk.
Further reading: Just Say "No" To Mud -- Philip Keyes.