The fastest, and cheapest method of underarm hair removal is to simply shave the area. This can be done in the privacy of your own home with something as simple as a disposable razor. However, it needs to be done on a regular basis so there is no re-growth. Frequently, when this method is used, the hair that does grow is thick and dark. So if you choose to use this method you need to make a routine and just shave every time you shower.
Another method of underarm hair removal is the use of depilatory creams. These are more expensive than the razor method, and since the underarm area is sensitive you must do a test patch first. Although the hair re-growth is a little slower than with the razor, you still need to use them fairly frequently, one to two times a week.
A third well used method for removing unwanted hair is waxing. In this process warm wax is spread over the area from which hair is to be removed, a cotton muslin cloth is pressed over the waxed area and then it is pulled off very fast. Needless to say there is some discomfort involved, but it decreases significantly after the first few times. This method takes a bit of practice initially so you might want have it done by a professional the first few times, just to get the hang of it. Over time waxing reduces the hair growth and the hair that does grow is usually softer and lighter.
The last two methods require a professional to do the job and are considerably more expensive than any of other methods. Electrolysis makes use of electric current to destroy the hair follicle. It takes a lot of time because each follicle has to be treated one at a time. Several treatments are required to get rid of the hair and is quiet uncomfortable. Finally there is the laser hair removal which destroys the hair in the follicle. Again it takes time, can be uncomfortable and does not work on light hairs.