When are aircraft required to dump fuel for emergency landings?
For many medium and large sized jets the maximum gross takeoff weight is higher than the maximum landing weight.  If the airplane has an emergency that requires an air return or other landing in the early part of flight, it is very likely overweight for landing.  The plane has 3 options at this point:


Land overweight
Dump fuel (if able)
Fly around at low altitude and high power to burn fuel


For some airplanes, option 2 is not available.  This includes airplanes that are not equipped to dump fuel (not all airplanes have this capability or have the capability under MEL) or are not approved to dump fuel at their current location and altitude.  For these airplanes, only option 1 and 3 are available.  

The choice between option 1 and 2/3 comes down to what kind of emergency it is.  If you have to be on the ground now, you won't care about being overweight, and will just put it down.  If you have time and landing now isn't a priority, then you would choose to dump or burn fuel.

If an airplane chooses to dump, they will generally choose one of two values of remaining fuel:


fuel such that they land at the maximum landing weight
absolute minimum safe fuel to land


This choice comes down to how likely the landing will end in an accident.  If you are sure the landing will be a non-event (e.g. air return for anti-ice failure) then you'll likely just dump until landing wont be overweight.  Conversely, if you have a control issue or something that may end badly, you want minimum fuel to lessen the potential for fire.  In these cases the ARFF crew will spray the runway with foam to assist in reducing the fire potential.