The most common causes for armpit rashes are allergic reactions to deoderants, soaps, detergents, or other similar products that irritate the skin. Shaving can also cause a rash.
If its a whitish rash,it could be a fungal infection. A reddish painful rash could be due to some inflammatory process secondary to infection-most likely involving the axilary hair roots for eg. folliculitis. Very rarely a hyperpigmented patch in the area could be suggestive of some internal malignancy[acanthosis nigricans].
A rash in the armpit area could be caused by a number of things, including sahving too closely, friction from tight clothes, or sensitivity to deodorant, soap, or detergent.
If you perspire or sweat quite regularly there and the area is not kept clean, your skin can become damaged from the constant moisture and cause possibly a fungal rash.