These are called <em>comparability graphs</em>.  A forbidden subgraph characterization was given by Gallai in 1967.  See the <a href="https://en.wikipedia.org/wiki/Comparability_graph">Wikipedia article</a>.

You can find the list of forbidden graphs in <a href="https://people.math.gatech.edu/~trotter/math-3012/12-Cover_Graphs_and_Comparability_Graphs.pdf">these slides </a> on Tom Trotter's page.  I think at one point he also had some course notes posted on it; it's also in his book on dimension theory of posets.  See also <a href="https://cstheory.stackexchange.com/questions/7677/is-there-a-list-of-forbidden-subgraphs-for-comparability-graphs">this post</a> on CSTheory stackexchange.