"Despite" means the same thing as "in spite of." It indicates a kind of contradiction or opposition, something that goes against something else. It's hard to give a clear definition, but here are examples of how it's used: "We went out despite the rain."/"We went out in spite of the rain." "I liked the movie despite the bad acting."/"I liked the movie in spite of the bad acting."
"Respite" is not the same as the other two. A "respite" is a rest or a break.