Let $A=\mathbb R[X,Y]$. Is it easy to classify the $\operatorname{Spec}A$? I guess it contains at least $(0)$ and $(p)$ for primes $p\in A$ but maybe some else sets. Is it easy to classify those?

Edit. Yes. I forgot maximal ideals. I think those are given in http://mathhelpforum.com/advanced-algebra/68149-maximal-ideal-r-x-y.html I'm not sure but I guess those can be written as $(p,q)$ where $p=X-a$ and $q=Y-b$ for $a,b\in \mathbb R$ or $p$ is linear and $q$ is $X^2+xX+d$ or $Y^2+cY+d$ where $c,d\in\mathbb R$ and $c^2<4d$.