An improper belief is a belief that isn't concerned with describing the territory. May be caused by cognitive biases. A proper belief, on the other hand, requires observations, gets updated upon encountering new evidence, and provides practical benefit in anticipated experience.
Note that the fact that a belief just happens to be true doesn't mean you're right to have it. If you buy a lottery ticket, certain that it's a winning ticket (for no reason), and it happens to be, believing that was still a mistake.