To hanker for something is to crave it, or really want it. On a hot, sunny afternoon, you might hanker for a tall glass of sweet iced tea.
People often hanker for relatively meaningless things, like chocolate cake or a nap or a new pair of shoes, but you can also hanker after more important things, like a long-lost friend or a new job. If you long for it, you hanker for it. The word's origin is a little fuzzy, but one good guess traces it to the Middle Dutch hangen, "to hang," from the idea of "hanging around" or "lingering."