At some time or another, the majority of people get that painful, red mass in one or both of their arm pits. This is called an abscess and it came about because a sweat gland got blocked or it is a boil, which is an abscess infection that forms around a hair follicle.
Common causes are your deodorant plugging your pores (consider changing your deodorant), extra heat generated that opens the pores and something gets clogged in a pore, or it simply can be an infection from bacteria build-up in the area. You can still exercise while you have this problem, but don’t think it will go away on its own.