Emma - I think it has all sorts of meanings and interpretations, but for me it's being able to move around and take control of positions and alter where you are in the room / how much you are moving about to get the baby out as comfortably (!) as possible.
So that's things like water birth, walking around, lying on your side or squatting, leaning over on all fours and being as active as possible to work through labour and the contractions. It's more difficult to have an acive birth in hospital because they are more likely to want to monitor the baby's heart beat / keep you on the bed as much as possible so that the MWs can examine you etc. You are also much more likely to have an epidural which basically means you are confined to the bed and can't push as easily as it's difficult to feel when to push. You are more likely to have forceps or ventouse which means you have to be on your back which means you are working against gravity (most women, given the choice, prefer to squat to give birth as this is the easiest way to work with gravity and getting the baby out).
But this is all from a book - I have not given birth so I am probably talking out my ass. I'm sure lots of ladies have successful active births at hospital as well as at home, but I think it's been proven that more intervention does take place at a hospital.