If you're interested in something, you like it and want to know more. If you're snoring through a movie, you're probably not that interested in it.
If you're interested in something, it has your attention. Some people are interested in sports; others are interested in books. If you like something and want to learn more about it, or just enjoy thinking about it, then you're interested. People can also be interested in things they have a stake in: we would say both the workers and employers are "interested parties" in a labor dispute, for example.