As its name implies, the average basis method requires you to compute an average basis for the shares that you are selling.
In many cases, your average basis is computed for you by your mutual fund and reported to you on a statement you receive around tax time. (The IRS, by the way, does not get this information from the fund.)
If it's not, add up your total basis in all the shares you owned before the sale, and divide by the number of shares. The result is the per-share basis. You can compute one average basis for all your shares—those held long-term and short-term ("single category method")– or compute the basis of shares held long-term and short-term separately ("double category method"). The single category method is a lot easier.
Using the average basis method is usually the easiest way to handle mutual fund sales, particularly if your fund keeps track of the number for you. Using it does give up some flexibility afforded by the specific identification method discussed next. That may or may not be important to you. Also, note this: once you use the average basis method, you need to use it for all future sales of shares in that mutual fund.