Periapical Tooth Abscess:  Here, decay has entered the pulp chamber (chamber in the tooth's center where the main nerves and other tissues live).  This infects the nerve area and it becomes inflamed and eventually dies. Bacteria take up residence in the pulp chamber and canals and live on the dead tissue.  The infection produces toxins, blood, gases, etc. and these impinge on the area around the root of the tooth, pushing on the surrounding bone.  The infection looks like a little pocket around the end of the tooth on an x-ray.  This infected area can be very large or quite small.
Decay is not the only thing that can kill a tooth.  Other causes include cracks in the tooth, trauma like falls and auto accidents (Wear a seatbelt) and sometimes, orthodontic treatment.

After Root Canal: In the bottom drawing we see the tooth after its pulp has been cleaned out and a root canal preformed.  A root canal fills the space with cement and plastic material so that bacteria are evicted.  Note that the area in the bone where the pus  pocket formed is healing with new bone. A filling has been placed in the top of the tooth.
For root canal information check out  Root Canal.