There are basically two types of extractions; simple or surgical.
A simple extraction is the type where there is no extra bone or tissue impinging on the tooth. These teeth can be removed by slightly elevating (moving) the tooth and removing it with a forceps.
The other types are surgical extractions which you commonly see with wisdom teeth (which are the 3rd molar at the back of the jaw). A partially impacted wisdom tooth is when the tooth is partially covered with bone or tissue. Usually you can see a part of the tooth visually in the mouth. The most accurate way to diagnose this is with an x ray of the area.
A fully impacted wisdom tooth is where there is tissue and or bone surrounding the entire tooth. These teeth need to have the tissue and bone removed surgically so that access to the tooth is possible in order to remove it. This type of extraction is usually referred to the Oral surgeon for removal.
Decayed or badly broken down teeth sometimes require surgical removal because there may be a need to expose more of the tooth in order to remove the tooth properly.