Flabby gum tissue sitting next to, or over, the back of a partially-erupted tooth can collect bacteria, food and debris under it and an abscess/infection can form.  Typically,  the patient cannot open his or her mouth wide and the pain may travel down into the lymph nodes of the neck.  The lower wisdom teeth are effected much more frequently than the uppers.

A dentist will carefully lift the tissue behind the tooth and clean the area out.  Sometimes a drop of pus and blood will come out when the flap is lifted.  This can afford relief, but usually an antibiotic is added as well. (Amoxicillin 500 mg, or erythromycin in case of amoxicillin/penicillin allergy.)  The sure cure is to have the tooth removed, and this is usually done by an oral surgeon. 
Often teens put the wisdom tooth surgery off and are treated more than once for this infection before they finally get the message and call the surgeon. It tends to happen at exam time when stress dampens the immune system a bit.  Some of my teen patients have even figured out how to clean under the flap of tissue that covers the back of an erupting lower wisdom tooth themselves with safety pins and various other household tools, but it is not so easy to get down around the side of the tooth where the pus and debris are trapped.

There are studies that suggest that the non-steroidal anti inflammatory drugs (i.e., Advil, Motrin, Ibuprofen family)  are best for dental pain.  If you buy Advil or the store brand Ibuprofen over the counter, it will come in 200 mg tablets and the directions on the box will tell you to take two every 4-6 hours.  In fact,  this is a low dose and it is more effective to take four tabs at once (800 mg).  This is the dose you would get if you were given a prescription for Ibuprofen.  Acetaminophen (Tylenol) does not seem to be very effective for dental pain in my opinion.

Sometimes narcotics are used for dental pain but studies have shown that ibuprofen, a non-narcotic,  is as effective as Aspirin with codeine or acetaminophen and codeine.  Codeine is a controlled substance.  Another controlled substance, oxycodone, may be prescribed for severe pain.  You need to have a prescription in hand at the pharmacy for this one.  My own feeling, is that I would rather get through a pain situation with non-narcotics if possible.  Taking too many at once can cause severe vomiting and they can leave you quite constipated for days.