How do Air Traffic Controllers determine which planes take off in a certain order?
The busier airports around the world are slot controlled, meaning that each flight is assigned a designated departure or arrival time that they must be ready for. In this case it’s pretty straightforward who goes first - whoever has the first slot. 

Non-slot controlled airports are usually on a first come first served basis, with some exceptions. Private flights have to wait for scheduled airline flights, and airliners wait for air ambulances. If two aircraft are departing on the same route, the faster one will go first if they have a significant speed difference. And the heaviest aircraft need greater separation for wake turbulence, so an A320 might get preference over an A380. 

That doesn’t mean your A380 will have to wait for 15 A320s though. In practice these are only guidelines and the decision as to who goes first is based on the needs of the traffic system on the whole considering all of the factors at the time.