A problem is complete for a complexity class if (1) it's in the class, and (2) everything in the class can be reduced to it (under some notion of reduction). So, if you can solve the complete problems for some class, then you can solve every problem in the class. The complete problems are the hardest.
In computational complexity theory, a computational problem is complete for a complexity class when it is, in a formal sense, one of the "hardest" or "most expressive" problems in the complexity class. Complexity classes are sets of all of the problems that can possibly be solved with at most a certain amount of some computational resource, and may include problems that actually require far less resources. The complete problems, however, are the most resource-intensive problems in the class.