Understanding the Gibbs measure seems to be essential to fulling understanding statistical mechanics. However, I don't know of any mathematically rigorous textbooks/lecture notes that talk about such subject. What are some good references that go into detail in the math and also provide examples of application in physics?
The basic reference for Gibbs measures and phase transition is Georgii's book. It covers a lot of topics on the subject. Other good (book) references are Velenik and Friedli's book and Sinai's book. These, however, are not totally dedicated to Gibbs measures, also covering various statistical mechanics topics.
In addition to the excellent references mentioned by Willy.K, you can find links to lectures by Adams, Disertori and Kupiainen on rigorous statistical mechanics on one of my old course webpages. Also, Roland Bauerschmidt has a very nice set of notes on spin models on this course page.