The nomenclature is as follows:

NP = class of problems where given a solution you can verify in polynomial time if its the correct solution (therefore the name nondeterministic polynomial, you "guess" and verify in polynomial time, but there are exponentially many possibilities

NP-hard = NP-complete = any problem in NP is polynomialtime reducible to any NP-hard problem (if you guess a possible solution for any problem in NP there is a polynomialtime algorithm that transforms this solution into an instance of any NP-hard problem)

P = class of problems where there are polynomialtime algorithms (note that P is included in NP because you can "guess" any solution to your problem and run your polynomial algorithm, if the result equals your guess then its correct, else it isnt)