What are the elements in this group? I think I am just getting thrown off by notation. I am not sure what the following means.
What is the group $G = (\mathbb{Z}/12\mathbb{Z})^2$? (What are the elements?)

I know the elements of $(\mathbb{Z}/12\mathbb{Z})$ are just $\{\bar0, \bar1, \bar2, \bar3, ... ,\bar{11}\}$. Would the elements of $G = (\mathbb{Z}/12\mathbb{Z})^2$ simply be $\{\bar0^2, \bar1^2, \bar2^2, \bar3^2, ... ,\bar{11}^2\}$?...but then those numbers mod 12?
$G=H^2$ means the direct product $G=H\times H$, so the elements are $(h,k)$ with $h,k\in H$. Now take $H=\mathbb{Z}/12$. If $H$ has $m$ elements, then $H\times H$ has $m^2$ elements.
Yes you're right. It means the direct product. But keep the modulo 12 in mind while computing.