Directed hypergraphs are used to model chemical reaction networks. This is closely related to the biological application Peter Arndt mentions in his answer.
The reaction network and the underlying hypergraph are related via the stoichiometry matrix, which is a matrix consisting of ones, zeros and minus ones which generalizes the adjacency matrix of a graph.
One obvious question you might ask about such a network is "are there any feedback loops"? This translates into the mathematical problem of finding directed hypercircuits in a directed hypergraph. This turns out to be an NP-complete problem (as shown in the 2008 paper On finding hypercycles in chemical reaction networks by Can Özturan) and so gives another example of the type Gowers mentions in his comment.
