WSJ article (subscription may be required) about everything you wanted to know (and more!) about deodorant and body odor. This is why we smell:
"Sweat itself is nearly odorless, but a particular type of sweat gland -- known as the apocrine gland -- found in hairy regions also secretes a milky fluid laden with protein. Bacteria catalyze an enzymatic reaction to break these secretions into food. Among the byproducts of this reaction are odorous compounds that cling so strongly to undershirts that they can survive a wash-and-dry cycle."
Yuck. This is how deodorants work so that we don't smell:
"All deodorants and antiperspirants combat odor in much the same way: There's an agent that kills the odor-causing bacteria that thrive in the warm, hospitable climes of the underarms, and a fragrance that covers up whatever odor sneaks out. To block sweat, antiperspirants contain aluminum salts that plug up sweat glands and reduce the flow of moisture. (In the U.S., about two-thirds of men prefer antiperspirants to deodorants. Nearly all women opt for antiperspirants.)"
Here's a fun fact, "emotional stress is an extremely important determinant of odor -- more so in many people than heat or exercise". Meaning that you may sweat more at the prospect of being laid off than working out at the gym.