A graph property is *hereditary* if it is closed under taking induced subgraphs (equivalently, if it is closed under removing vertices). A graph property is *monotone* if it is closed under taking subgraphs. (Note that "monotone" is sometimes used in different ways from what I've just written.) Thus, every monotone property is hereditary (but not conversely). Every monotone property can be characterized by a set of forbidden subgraphs, and every hereditary property can be characterized by a set of forbidden induced subgraphs. (In each case, the set of forbidden graphs may be infinite.)

Given a (monotone or hereditary) property $\mathcal{P}$, it's easy to define a set $\mathcal{F}$ of forbidden subgraphs (or forbidden induced subgraphs): if $\mathcal{U}$ is the set of all finite unlabeled graphs, set $\mathcal{F} = \mathcal{U} \setminus \mathcal{P}$. However, this is not very useful. Many hereditary properties can be characterized by simpler set of forbidden subgraphs: for example, perfect graphs are exactly the graphs with no induced $C_{2k+1}$ or $\overline{C_{2k+1}}$ for any $k \geq 2$.

However, I'm having a harder time coming up with examples of monotone properties that have "non-trivial" characterizations in terms of forbidden subgraphs. Of course, for any graph $H$, the class of $H$-free graphs is trivially characterized by taking $\mathcal{F} = \{H\}$. The only other example that comes to mind is bipartite graphs, which are characterized by forbidding odd cycles.

What other examples are there?