How can you prove that the square root of two is irrational? I have read a few proofs that $\sqrt{2}$ is irrational.

I have never, however, been able to really grasp what they were talking about.

Is there a simplified proof that $\sqrt{2}$ is irrational?
You use a proof by contradiction. Basically, you suppose that $\sqrt{2}$ can be written as $p/q$. Then you know that $2q^2 = p^2$. As squares of integers, both $q^2$ and $p^2$ have an even number of factors of two. $2q^2$ has an odd number of factors of 2, which means it can't be equal to $p^2$.
(Since p and q are coprime integers it is supposed to be in lowest form. But we demonstrated the existence of a fraction having lower positive integral denominator.)
It is proof by Contradiction