What do prime ideals in $k[x,y]$ look like?

For $k[x, y]$ this is not as bad as it sounds! The saving grace here is that $k[x]$ is a PID.

Proposition: Let $R$ be a PID. The prime ideals of $R[y]$ are precisely the ideals of the following form:

• $(0)$,
• $(f(y))$ where $f$ is an irreducible polynomial (recall that Gauss' lemma is valid over a UFD, so irreducibility over $R$ is equivalent to irreducibility over $\text{Frac}(R)$),
• $(p, f(y))$ where $p \in R$ is prime and $f(y)$ is irreducible in $(R/p)[y]$.

This is a nice exercise. If you get stuck, I prove it in in this blog post. The primes of the third type are maximal, so when $R = k[x]$ you've already listed them (by the weak Nullstellensatz). The only new prime ideals are those of the second type; they correspond to irreducible subvarieties of dimension $1$.

In general I'm not even sure what would count as a reasonable description, and I don't know enough algebraic geometry to comment.

The prime ideals of $k[x,y]$ are $0$, the maximal ones, and $(P)$ where $P$ is any irreducible polynomial. This is because $k[x,y]$ has dimension two, and is a UFD. For higher-dimensional rings things are more complicated, and there is no explicit answer. However, many things can be said, for example about the minimal number of generators for prime ideals. Good references are Introduction to Commutative Algebra (Atiyah-Macdonald), Commutative algebra (Matsumura), Commutative algebra with a view toward algebraic geometry (Eisenbud). The last one is more complete and self-contained.

For $k[x,y]$ there are excellent answers here; however, if one still wonders what happens in $k[x_1,\ldots,x_n]$, $n>2$ (as was commented), then it is worth to mention Theorem 2.6 of Miguel Ferrero's paper, which deals with prime ideals in polynomial rings in several variables, $n \geq 2$.