You would use the FOIL method when you are trying to multiply two binomials together...that is, an expression in the form (x + A)(x + B).
The term FOIL stands for First Outer Inner Last (sometimes also using Outside and Inside instead). This refers to the order of multiplying terms together.
I will explain it through an example, which should hopefully be all that is necessary. Let's try to find the product of (x+4)(x+6).
To apply the FOIL method, you have to first identify and understand what terms we will be talking about.
Here is what the letters in FOIL mean:
The term Outer means "the outer terms of the product expression." In this case, the first x and 6 are the Outer terms.
The term Inner means "the inner terms of the product expression." In this case, the 4 and the second x are the Inner terms.
The term Last means "the last terms in each binomial." In this case, 4 and 6 are the Last terms.
So then, now that you have identified each of the components of FOIL, now all you do is MULTIPLY the two terms for each letter of FOIL, and then add them up (there will be 4 products).
For this example, we'd have these products:
F: (x)(x) = x2
O: (x)(6) = 6x
I: (4)(x) = 4x
L: (4)(6) = 24
Now, we just add them together, combining like terms wherever possible (here, the inner and outer terms):
x2 + 6x + 4x + 24
x2 + 10x + 24
And that's all there is to the FOIL method. If you'd like more examples or a better explanation, leave me a comment. But I think that the example should be sufficient for now to demonstrate the concept.