When it comes to recommendation systems and natural language processing, data that can be modeled as a multinomial or as a vector of counts is ubiquitous. For example if there are 2 possible user-generated ratings (like and dislike), then each item is represented as a vector of 2 counts. In a higher dimensional case, each document may be expressed as a count of words, and the vector size is large enough to encompass all the important words in that corpus of documents. The Dirichlet distribution is one of the basic probability distributions for describing this type of data.
The Dirichlet distribution is surprisingly expressive on its own, but it can also be used as a building block for even more powerful and deep models such as mixtures and topic models.