Which schemes are divisors of an abelian variety?

Any curve of genus greater than two, whose Jacobian $J$ is simple, will do. If it were a divisor on an abelian surface $S$, then there would be a surjection $J\to S$ with positive dimensional kernel, contradicting the simplicity of $J$. Most curves of genus larger than two have this property; a randomly chosen example is $y^3 = x^4 - x$.

An obvious class of counterexamples are uniruled varieties. In fact, abelian varieties contain no rational curves.

More generally, and for the same reason, if $X$ is any algebraic variety that contain a (possibly singular) rational curve, then $X$ is not a subvariety of an abelian variety, in particular it is not a divisor there.

Here's another answer using the Albanese that's of a slightly different flavor. Let $X$ be $n$-dimensional and suppose that $h^0(X,\Omega^1_X)<n$. Then any map $X\rightarrow A$ where $A$ is an abelian variety factors through the Albanese, which is of dimension less than $n$, so $X$ can't be a divisor on any abelian variety. So as an example you could take any simply connected variety. Of course, $\mathbb{P}^1$ does the trick.