Peeling around the toenails. Do I have athletes foot? What else gives you peeling?

Most like fungus. It is probably a fungal infection. Use a good topical antifungal as labeled, I like Lamisil (terbinafine), but most will work. If this does not improve in a week or two or gets worse, see your podiatrist or dermatologist for an evaluation. Dr. L.
Several things. You may have an athletes foot infection, but it could be a bacgterial infection. Should be seen by a doctor to determine specific cause.