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NP vs NP-Complete vs NP-Hard — what does it all mean?

Euler circuit problem can be easily solved in polynomial time. Hamilton circuit problem is proved to be NP-hard. Nobody in the world can give a polynomial time algorithm for a NP-hard problem.

What is meant by polynomial time and NP-hard? I know what is O(n).

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Polynomial time means that there exist a constant a, such that the complexity of your algorithm is O(n^a).

Here is an explanation about NP-hard.

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it is really difficult to understand NP-hard
For a gentler introduction, see P versus NP Problem

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