FIFO --> First In, First Out --> the assumption that goods that arrived first will also be the first to be sold. For example, 1 July 2008 a supermarket buy 10 boxes of milk @ $5, with an expiry date of 1 January 2009. 1 September 2008, it then buy again 10 boxes of milk @ $6, with an expiry date of 1 March 2009. It is very logical for the supermarket to sell the first 10 boxes of milk, before selling the second ones, perhaps by displaying the first 10 boxes in the front row.
Suppose that in a month, the supermarket sold 5 boxes of milk. How much is the cost of the 5 boxes of milk sold that month (in short: how much is the Cost of Sales)? That will be $5 x 5 boxes = $25

LIFO --> Last In, First Out --> the assumption that goods that arrived last will also be the first to be sold. For example, using the supermarket data above, the Cost of Sales will be $6 x 5 boxes = $30

Average --> not really an assumption, but a decision that the company will average the costs of purchases. For example, using the supermarket data above, the Cost of Sales will be $5.5 x 5 boxes = $27.5

Which ones to use in your company?

Contrary to the teachings of the American text-books that all three are just assumptions and have nothing to do with the actual (physical) goods movement, I recommend that companies use the assumption that most closely resemble their goods movement. That way, your cost of inventory, cost of sales, and subsequently gross margin, most truly reflect the actual economic performance. Under no circumstances shall a company choose an assumption based on the effect it has on the cost of sales and gross margin.

Another aspect to consider is the cost vs benefit of implementation within the company's information system. Implementing FIFO and even more so LIFO, is more difficult and more restricting than Average. For simplicity and flexibility, and if you are not aware of the complications resulting from FIFO and LIFO assumptions, it is better to just use Average.