This may serve as a cheat-sheet for someone who wants to get an understanding of the computational complexity terms, hopefully in the shortest possible time.
P – Problems that can be solved in polynomial time, obvious to this is, solutions to such problems can be verified in polynomial time as well. Example – Sorting problem.
NP (Non-Deterministic Polynomial) – Problems that can be verified in polynomial time but not necessarily can be solved in polynomial time. Example – Subset Sum Problem.
NP Complete – Problems that are as hard as any problem in NP. These are problems on the edge of NP space.
NP Hard – Problems that are higher in complexity than that of NP Complete problems. Example – Travelling Salesman Problem (TSP).
EXP – Problems that can be solved in exponential time. While TSP is a N! problem, it can be solved in exponential time using Dynamic Programming with 2^n*n^2.
P!=NP means, problems that can be verified in polynomial time can’t necessarily be solved in polynomial time.