For something like factoring a^2 - b^2 it is probably best simply to memorize them. Likely ones to come up are:

a^2 - b^2 = (a + b)(a - b)

a^3 + b^3 = (a + b)(a^2 - ab + b^2)

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

a^2 + 2ab + b^2 = (a + b)^2

a^2 - 2ab + b^2 = (a - b)^2

Then there are the ones that use the distributive property in reverse. These are always of the form:

ab + ac = a(b + c)

For example, consider 3x^3 - 27xy^2. What is common between the two of them? The factor 3x. So

3x^3 - 27xy = (3x)(x^2 - 9y^2)

But we aren't done yet! x^2 - 9y^2 = (x)^2 - (3y)^2 = (x + 3y)(x - 3y) so we have:

3x^3 - 27xy = (3x)(x^2 - 9y^2) = (3x)(x + 3y)(x - 3y)

I don't have time right now to go through factoring trinomials. Perhaps someone else will. If not I'll get back to you later.

-Dan