Most body odor is the result of bacterial breakdown of sweat.
Perspiration itself is normally odorless. An easy way to reduce the amount of bacteria under the arms is to splash on rubbing alcohol. Most commercial deodorant products contain irritating or harmful ingredients, including aluminum salts and dyes. Avoid the antiperspirant varieties. You can find better products in health-food stores, such as those containing extracts of green tea, which is antibacterial.