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, a 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.