Terminal Control is often referred to as TRACON (Terminal Radar Approach Control) and contains both approach and departure systems. Since TRACON controls both sections, I will split the sections.

Taken from wikipedia
Departure:
Departure aircraft are received from the tower and are generally 1,000 feet to 2,000 feet high, climbing to a pre-determined altitude. The Terminal Control Center controller working this traffic is responsible for clearing all other Terminal Control Center traffic and, based on the route of flight, placing the departing aircraft on a track and in a geographical location (sometimes referred to as a "gate" or "exit"). This positioning is designed to allow the en-route center to integrate the aircraft into its traffic flow easily.

Approach:
Arrival aircraft are received from the en-route center in compliance with pre-determined agreements on routing, altitude, speed, spacing, etc. The Terminal Control Center controller working this traffic will take control of the aircraft and blend it with other aircraft entering the center airspace from other areas or "gates" into a single, parallel or perpendicular final for the runway. The spacing is critical to ensure the aircraft can land and clear the runway prior to the next aircraft touching down on the runway. The tower may also request expanded spacing between aircraft to allow aircraft to depart or to cross the runway in use.

What the above definitions mean to you is:
1. It is TRACON's job to maintain a safe flow of traffic into and out of the surrounding airspace.
2. Approach controllers generally take aircraft from around 60 miles out and vector them onto an intercept course for a runway's final.
3: Departure controllers take over around 1,000 agl and vector aircraft safely out of the airspace.

Time for more radio samples!
Generally speaking, an aircraft is handed over to Departure by the Tower controller. Basically, the initial call up will just be to acknowledge their contact.
P: Detroit Departure, (callsign) with you climbing through (current altitude) for (assigned altitude).
C: (callsign), Detroit Departure with you, good evening. Radar contact.

You will then issue vectors as necessary to get the aircraft clear of your airspace. As a Departure controller, you should be familiar with all of the airports SID(Standard Instrumant Departure) procedures. These are the "gates" or "exits" that were mentioned earlier. They are designed to make your life easier by ensuring that all aircraft are on the same page.

The following is a sample of an Approach controller's initial contact call.
P: Detroit approach, (callsign) with you at (altitude), good evening.
C: (callsign), Detroit Approach with you, good evening. Radar contact. Expect vectors to the ILS approach runway (xx) current altimeter (xx.xx)

You will then issue vectors to get the aircraft down to the altitude and course he/she needs to be to intercept the runway final. Once they are on an intercept course, you can turn them over to Tower(usually around 15-20 miles out). As an Approach controller, you should be familiar with all of the airports STAR(Standard Terminal Arrival) procedures. These are the "gates" or "entrances" into the airspace, that were mentioned earlier. They are designed to assist you in the safe flow of traffic to a runway's intercept course.

In closing, I am a firm believer in seat time as the predominant way of learning ATC. The above examples are merely an introduction to air traffic control procedures practiced at DVATC and are in know way intended to be the "gospel" on the subject. It is required that you are trained and certified by a DVATC ATC Instructor.