Rash under arms

It sounds like it could be an allergic reaction to something. Have you switched deoderants, soaps, laundry detergents?

To figure out what it going on, try using a laundry detergent with no fragerance, or even better use dreft (the detergent you use to wash infants clothing because their skin is so sensitive). If you have started using a new soap or deoderant, go back to your old one or buy hypoallogenic, non-fragranced products.

For the discomfort, take an antihistamine and get some anti-itch cream (Eucedrene, Gold Bond, etc)

If this doesn't work, then you may want to see a dermatologist. There are so many reasons and kinds of rashes, it's herd to say, but it sounds like hives or a reaction to something on the skin to me.