Since every emergency is a unique experience for both aircrew and ATC, then the way it is dealt with is unique.
Whilst the crew have the checklists to go through, the I'll do this, if it does this then we do this etc etc. In ATC things are a bit different, our priority is to get you to the nearest suitable airfield and on the ground as quickly as possible (if that means routeing you out to sea to dump fuel, then we will, if you need to dump fuel asap then we'll climb you or in worst case scenario, cover the residents of the local town in Jet A1).
There are certain details that we will need (for the emergency services and for our own benefit) e.g how the aircraft performance is affected i.e you've lost and engine (which one?), can you only turn in a certain direction? are any other engines or systems affected? How will that affect the approach (will it be a stable approach, a faster than normal landing?)
How many people on board the aircraft? (although I believe that in the UK the emergency crews regard each emergency a/c as being full to capacity).
Can the aircraft maintain height? do the crew require unrestricted descent? How many miles to touchdown do they require? etc etc etc.
The list is endless, basically we do what we can to get the a/c on terra firma as soon as is practicable.