There are two main methods of getting the hair of your underarms permanently removed: laser hair removal and electrolysis. They operate in different ways, but the end goal is the same -- to remove hair and keep it from ever growing back. You don't have to worry about razor burn when a razor never touches your underarms.
Electrolysis destroys the hair follicles (the part of your skin that hairs grow from) via the insertion of a tiny electrode. The electrode zaps the follicle, which will never grow another hair again. It's a tedious process, but is generally regarded as an effective method of removing hair permanently.
Laser hair removal takes advantage of the fact that the pigments in dark hair follicles are darker than the surrounding skin. The follicle absorbs more light and heat than the skin, so a targeted laser can destroy the follicle without harming surrounding skin cells. However, because it relies on the contrast between the hair follicle's color and skin color, it works best with dark hair and light skin. While it can work with dark skin, people with blonde or light red underarm hair will need to find another method of underarm hair removal.
Both electrolysis and laser hair removal have varying degrees of success, depending largely on the skill of the operator. They are both also quite expensive. A single session can cost from $100 to several hundred dollars, and seven or more sessions may be needed to complete the treatment.