dermatitis (atopic, contact, allergic),
heat rash, excessive sweating, folliculitis,
boils, scabies, psoriasis, some infection leading to enlargement of axillary lymph nodes. Sometimes even breast cancer or hodgins disease may present with enlarged axillary lymph nodes.
You should wash the area with hypoallergic antibacterial soap followed by rinsing thoroughly, apply moisturizer, wear loose fitting clothes and avoid deodorant / cosmetics for a couple of days.