EDIT: This seems to be a similar question (assuming the question is about the history, not the "hardness"):
Stack Overflow - First NP complete problems
Turing machines are important because they are considered a model of computation that applies to essentially any computer (except perhaps quantum computers) http://en.wikipedia.org/wiki/Church Turing_thesis so they help us study computation.
It's been a while since I studied complexity theory but I'll try explaining in more detail:
NP is non-deterministic polynomial time. That is the problem can be solved in polynomial time using a non-deterministic Turing machine. Non-determinism in this statement implies the turing machine can explore different solutions concurrently.
The concept of "time" is the number of computation steps as a function of the length of the input. (as the input length goes to infinity and ignoring any constant factors).
You'll need to look at the model of a Turing machine to get a better idea of what is meant by the length of input and steps.
And so, there are many problems in complexity class NP, some of them really easy. Some of them "hard". They're hard because there is no known deterministic Turing machine, polynomial time, algorithm for solving them. They're also very interesting because if one of them can be solved in polynomial time on a determinisitic Turing machine then all of them can be.
When you do a formal proof for NP-completeness you need to show:
- That the problem is in NP. That is you show some non-determinsitic Turing machine that solved the problem.
- That if you can solve it in polynomial time on a determinstic Turing machine then you can solve some other NP-complete problem in polynomial time on a determinstic Turing machine. You do this via reduction - transforming one problem into another in a number of steps that's a polynomial function of the input. This shows that it's at least as hard to solve as the other problem.
I think I got that right :-) I hope it makes sense.