It sounds like fin rot.
This info was taken from Darkmoons Betta Fish Disease and Treatment Part 1 thread: http://www.bettafish.com/showthread.php?t=73332
Tail rot or fin rot
•Symptoms: Betta’s fins and/or tail seem to be getting shorter and shorter or they seem to be falling apart and dissolving, Black or red along the edge of the betta’s fins/tail, Bloody tips, Behavior may not change
Treatment: Conservative: Treat with Aq.Salt at 1 tsp/gal. Increase water changes to 100% daily. Replace accurate amount of salt following water changes. Add Stress Coat to help repair tissue. If there is little to no improvement within the first 5 days, you can increase the salt dosage gradually to 2tsp/gal but do not continue any salt treatments past 10 days. Medication: If Conservative treatment is ineffective use API Tetracycline, API Fungus Cure, API Triple Sulfa, OR API Erythromycin. Also add Stress Coat to help regrowth. Continue until fins/tail stop receding and start showing some new growth.
I'm slightly new when it comes to Bettas being sick so this is the best I can give you until another member comes along who knows more then me :)