Those are pretty complex questions, but actually happen to both fall under my areas of expertise. Here's a simple explanation:
Multiple Intelligence Theory is the idea that there's more than one kind of intelligence. So, basically, you can be musically intelligent or mathematically intelligent. There isn't just one overall intelligence to rate people by. You can be very intelligent in one area and not in another. The theory is, well... a little simplistic and outdated, if at least a good way to get people to broaden their minds about the concept of intelligence. So this theory falls more under the line of psychology.
Artificial intelligence is, in theory, the idea of a computer being able to "think." As in, not to just do what its specifically programmed to do, but to be able to reason out its own solutions, like a human. In practicality, that idea is way far away, so artificial intelligence just refers to the idea of trying to make "smart" computers. For instance, a computer that can beat a human at chess is a simplistic form of artificial intelligence. It's more of a spectrum than a single concept.
As for how it applies to games, specifically mmorpgs, it'd be used to make the NPCs seem more alive and real. Allowing them to seem more alive and be less predictable.