The fitting you see in the picture connecting the horizontal and vertical pipes is called a sanitary tee. The vertical pipe above the tee is the vent, and the pipe below is the drain.
The horizontal pipe is the trap arm. The trap arm connects to the trap, the trap connects to the tailpiece and the tailpiece connects to the sink.
Code limits the length of the tailpiece to no more than 24 inches. Not shown in the picture is a test tee, which has a threaded plug that allows you to access the drain for cleaning.
The test tee may be installed either above or below the sanitary tee. Kitchen drains usually clog where the trap arm meets the sanitary tee, so installing a test tee above makes it easy to remove any clogs.
You need to know the type and model of sink and the model of garbage disposal you are installing. Because most sinks are cabinet mounted, you want to know the depth of the sink bowl or bowls and the height of the outlet on the garbage disposal.
After you have these numbers you can determine the best height for the sanitary tee. 14 to 16 inches from the floor usually works well for most cabinet-mounted kitchen sinks.
Island sinks are a special problem because there is no place for the drain vent. To solve this problem, make a loop vent in the island.
(See the picture at left) The top of the loop should be as high as possible under the sink. Both legs of the loop attach to a drainage line under the floor, but one leg has a vent tee installed immediately under the floor and above the drainage line.
Attach your horizontal vent line to this tee and run it to the nearest vent. (You can see the vertical vent in the back of the photo) You need your horizontal vent pipe to rise 1/4 inch per foot from the vent tee to the vertical vent, and any horizontal drain pipe to fall 1/4 inch per foot.
Consult your local plumbing code for specific municipal regulations.