It has been some time since I left university...
In a not too formal language, an $n$-dimensional vector is an indexed set of numbers $\{i_1, ..., i_n\}$. A $n\times m$ matrix is a set of numbers with a two-dimensional index $\{i_{11},...,i_{n1},i_{m1},...,i_{mn}\}$.
What is the generalization of this, i.e. a set of numbers with an $n$-dimensional index? What is it called?