Tower controllers coordinates the sequencing of aircraft within 5 miles of the airport that are landing, as well as those that are just about to take off or have done so. Tower will also handle very low altitude aircraft that desire to pass within 5 miles or so of the airport, such as helicopter traffic.
Departure control will sequence aircraft who were just switched from the tower frequency away from the airport to their first or second en route fix (or few fixes), as well as begin climbing them out of their airspace and off to the next section of airspace, which could be another approach controller or a center controller. Center controllers most often controls the higher airspace from 10,000 feet or so up to 60,000 feet. Center will also handle lower altitude aircraft that are well outside of any towered airports, such as those landing or departing the numerous small, non-towered airports.
Approach control will control arriving aircraft within a ring of typically 30 miles to 5 miles outside the airport by sequencing them towards the airports and in line to land. Approach control will also handle arriving and departing aircraft that are using nearby, satellite airports. For example, Buffalo approach, who handles arriving traffic into Buffalo Airport in western NY state, will also handle instrument flight rules (IFR) aircraft that need to land or have just departed nearby untowered Dunkirk airport.