NP-hard - Definition
Here is a definition which describes the relationship between hard and complete:


  Given a set C of problems and a reduction R, it is natural to ask if there are problems Π ∈ C such that any problem Π′∈C,  R-reduces to Π. Such “maximal”
  problems are called in complexity theory complete problems. Let C be a class of
  problems and R be a reduction. A problem Π ∈ C is said to be C-complete (under R-reduction) if for any Π ∈′ C, Π′ ≤R Π. A C-complete problem (under reduction R) is
  then (in the sens of this reduction) a computationally hardest problem for class C. For instance, in the case of NP-completeness, NP-complete problems (under Karp-reduction) are the hardest problems of NP since if one could polynomially solve just one of them, then one would be able to solve in polynomial time any other problem in NP. Let C be a class of problems and R a reduction. A problem Π is said to be C-hard (under R-reduction)
  if for any Π ∈′ C, R Π≤ Π ′. In other words, a problem Π is C-complete if and only if Π ∈ C and Π is C-hard. 


References


An Overview on Polynomial Approximation of
NP-Hard Problems
Computers and Intractability: A Guide to the Theory of NP-Completeness