A contentious issue is one that people are likely to argue about, and a contentious person is someone who likes to argue or fight.
Some issues — like abortion, the death penalty, and gun control — are very controversial. They're also contentious, because people tend to argue about them, and the arguments will probably go on forever. Contentious issues get people angry and in a fighting mood. On the other hand, some people always seem to be in a fighting mood, no matter what the issue is. People like that are contentious too.