Bildungsroman is a big, scary, German word for something that is actually pretty familiar: a coming-of-age story. A bildungsroman charts a protagonist's development from childhood to adulthood, from innocence to experience, from being a total noob to pwning left and right. In short, bildungsroman is concerned with the education and progress of its main character.
Famous bildungsromans include James Joyce's The Portrait of the Artist as a Young Man and Charlotte Bronte's Jane Eyre. You'll also find lots of coming-of-age bildungsromans in the young adult genre, like in Avi's The True Confessions of Charlotte Doyle or S.E. Hinton's The Outsiders.