What is an NP-complete problem? Why is it such an important topic in computer science?
NP stands for Non-deterministic Polynomial time.
This means that the problem can be solved in Polynomial time using a Non-deterministic Turing machine (like a regular Turing machine but also including a non-deterministic "choice" function). Basically, a solution has to be testable in poly time. If that's the case, and a known NP problem can be solved using the given problem with modified input (an NP problem can be reduced to the given problem) then the problem is NP complete.
The main thing to take away from an NP-complete problem is that it cannot be solved in polynomial time in any known way. NP-Hard/NP-Complete is a way of showing that certain classes of problems are not solvable in realistic time.
Edit: As others have noted, there are often approximate solutions for NP-Complete problems. In this case, the approximate solution usually gives an approximation bound using special notation which tells us how close the approximation is.