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 $\dfrac{p}{q}$. 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. Therefore, $2q^2$ has an odd number of factors of $2$, which means it can't be equal to $p^2$.