Bipartite graphs model two-sided relationships:
people and courses: edges connect people to courses they are enrolled in.
Job applicants and positions: edges show who's qualified for what.
people and products: edges represent purchases or reviews.
Authors and papers: edges link authors to their publications.
In each case, edges only cross between the two types. Recognizing bipartite structure gives you access to special algorithms for matching and assignment that do not work on general graphs.