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I think of an array as a function from a finite set of ordered k-tuples of natural numbers to any other set S. So for example, if you let k=2 and S=R then you have your average matrix from baby linear algebra. The function gives you a rule such that given any (i,j), where i is less than some fixed n and j is less than some fixed m, you can find a real number Aij. You are essentially "naming" Aij with i and j.
When k>2 then it is appropriate to use the term array. If you want to get more technical look up indexing functions.
