Fortunately/unfortunately ATC, crews, companies etc all have slightly differing goals as to what they're trying to achieve, but ultimately, safety is at the top of everyone's list. Since ATC has the 'big picture' with regard to traffic flow, conflictions, priorities, etc, they are the ones who have to make it fit together through the various techniques and strategies that have already been explained here.
If you can do it, my suggestion would be to get on the phone to ATC and arrange for a visit to one of the centres and sit in with approach and/or the arrivals sectors one afternoon/evening during the week. That way you'll get to see the good, the bad, and the ugly of how things work from the ATC perspective.