So I am watching Lord of The Rings, and as great as a movie as they are I'm wondering to myself what an actual proper siege of a town or castle would look like.

During a war in England or Northern Europe (during the 12th or 13th century) during the middle ages how would one side try to take the others castle, how successful were these methods and how long did a siege normally last? I'm getting the impression that assaulting a stone wall might not be easy with only a sword or shield.