Fin Rot
Fin rot is actually one of the illnesses that I've dealt with (and cured!) in one of my fish. The instructions below give the short story, but you can see the complete description of my coping with fin rot at this blog post: Treating Fin Rot: Finally Done!
Symptoms: Betta’s fins and/or tail seem to be getting shorter and shorter, or they seem to be falling apart and dissolving. There may be a darker color (or a reddish one) to the edge of the betta’s fins/tail. He may be still active and eating normally, or may have stopped eating, fins may be clumped, color may be pale.
Treatment: Fin rot is caused by a bacteria, and so your first job is to kill the bacteria. Do a full water change. Use tetracycline or Ampicillin (maybe combined with Fungus Eliminator if you fear a very serious infection). Change water every third day and add a new dose of same medication, but if in doubt, just follow the directions on the box. I've had the most success with Tetracycline, which I bought at Petsmart. These medications will kill the rod bacteria that cause the fin rot. Continue until fins/tail stop receding (one way to tell is if the blackish edge goes away) and start showing some new growth. This may take up to 4 weeks, so don’t give up. I also suggest using Jungle Anti-Bacterial Pellets, which is an antibiotic food that helps the betta's little immune system fight the infection from the inside as well. Use these pellets exclusively as you treat fin rot.
Once the bacteria is killed, use Melafix, which encourages fin growth after loss from infection or injury (see the First Aid page for more on Melafix and its uses). Melafix does not kill the fin rot bacteria so this is why you should use it after the bacteria is gone. Over time you will begin to see new webbing in between frayed parts of the fin or tail. This new growth may look discolored but might gain color as time goes on.
If the fin rot is very persistent, it can eat away at the fins and tail and then attack the body. If the rot is not going away, begin using the tetracycline and the Ampicillin together, possibly doubling the dosages. This may be the only chance to heal this advanced fin and tail rot. When rot stops, use Melafix as described.