I can't believe nobody has yet mentioned the book [Period mappings and period domains][1] by Carlson, Müller-Stach and Peters. Chapter 1, the introduction, is written almost as a story, starting with the pure Hodge structure on the cohomology of a Riemann surface and introducing mixed Hodge structures by looking at degenerations. Further niceties await the reader in subsequent chapters.


  [1]: http://books.google.com/books?id=ps6WSWhdlQIC&lpg=PP1&dq=Carlson%2520Period%2520mappings&pg=PP1#v=onepage&q&f=false