What are NP-complete problems and why are they significant?

NP-complete problems are the hardest problems in NP (verifiable in polynomial time) - if any NP-complete problem has polynomial solution, all NP problems do (P=NP). Examples: SAT, Traveling Salesman, Graph Coloring, Knapsack (decision version). Proven NP-complete via reduction from known NP-complete problem. Significance: no known polynomial algorithms exist, likely impossible, so we use approximation algorithms, heuristics, or solve restricted cases.