What are NP-complete problems and why are they so important? I keep hearing questions about whether something is NP-complete, but they never really mention what it is. Why do people care so much about NP-complete problems?
A problem is in class NP if its solution may be verified in polynomial time, that is if the dimension of the problem is n you may be sure that for large enough n you need less than r&middot;nk operations to verify the solution. 

A problem is in class P if its solution may be found in polynomial time, instead. A problem in P is in NP by definition, but the converse may not be the case; probably the most important open question in computer science is whether classes P and NP are the same, that is P=NP. 

NP-complete is a family of NP problems for which you know that if one of them had a polynomial solution then everyone of them has. 

(EDITED) For the time being, only known algorithms for NP-complete problems are exponential in number of operations, so they are not practically solvable for n large.
A decision problem D is said to be NP-complete if
    1.it belongs to class NP;
    2.every problem in NP is polynomially reducable to D.

Reducable means we can convert one instance of problem A in to problem B[NP problem] then it means A is reducable to B.