Why we want to deal with NP completeness?

How can we achieve this goal?

Heuristic

approximate solution
What’s the fairly good solution (maximum or minimum) to some question. Approximation ratio is the key to describe how good an approximation is.
Brute-force methods
In exponential-time.
recursive backtracking
For 3-SAT problem.

What is the most important thought behind these dealing methods?

Any helpful things?

Additional information