For example, what is the difference between

$\displaystyle \mathbb{Z}_2[x]/(x^2 + 1)$ and $\displaystyle \mathbb{Z}_2[x]/(x^2 + x + 1)$?

What do they look like?

Is $\displaystyle \mathbb{Z}_2[x]/(x^2 + 1) \cong \mathbb{F}_2[x]/(x^2 + 1)$?

I'm having trouble "getting it" if that makes sense.