What 'War and Peace' is to the novel and 'Hamlet' is to the theater, Swan Lake' is to ballet - that is, the name which to many people stands for and sums up an art form.

This quote is saying that Swan Lake is a very significant ballet. It is essentially the definitive ballet for many people – in the way that War and Peace is the definitive novel and Hamlet is the definitive play. Swan Lake is the name that represents the art of ballet. It is essentially a symbol of the entire genre, being seen as the go-to example of what a successful ballet can be.