Two kinds of difficulty
Some problems are easy to solve; others are merely easy to check. The class P holds problems solvable quickly, in time bounded by a polynomial in the input size. The class NP holds problems whose proposed answers can be verified quickly, even if finding them seems hard.
What NP complete means
A problem is NP complete when it sits in NP and is also among the hardest in NP, meaning every other NP problem can be transformed into it efficiently. If anyone ever solved one NP complete problem quickly, every NP problem would fall too. Classic examples include boolean satisfiability, Hamiltonian path, and the traveling salesman decision.
The great open question
Nobody knows whether P equals NP. Most researchers believe they differ, so that NP complete problems have no fast algorithm. But this remains unproven, one of the deepest open questions in all of computing.
Why it matters in practice
Recognizing that a task is NP complete is liberating, not defeating. It tells you to stop hunting for a perfect fast algorithm and instead reach for approximation, heuristics, special cases, or smaller inputs.
Key idea
NP complete problems are the hardest in NP, easy to verify but with no known fast solver, and solving any one quickly would solve them all.