…or, for that matter, capping a new well?
My impression is that (in at least some cases), the oil is coming out of the well under pressure; is putting out the fire only part of the problem?
Also, is it common for oil wells to keep on producing under pressure, or is it always the case that fluid must be pumped in to maintain it?
More info here.
Putting out the fire AND putting a cap on the well - those are two things primarily. The fire is put out in many ways - lots of water / foam / Solid CO2 / explosives are used.
To your second question - most oil wells start out under pressure. Near the end of the life of an oil well the pressure drops and need external means to raise the pressure inside to remove the remaining oil.
This site goes into some of the background and extinguishing methods for blow outs and other oil well fires.
If you can find an Imax theatre in your area playing it, Fires of Kuwait is an excellent documentary which shows exactly how the oil well fires are extinguished.
Oh, I should add that the documentary is apparently available on video now as well, so check your local rental store.