If you're familiar with quadratic forms in multidimensional calculus you'll know that any bilinear relationship f(x, y) on two vectors from two vector spaces x and y can be represented by a matrix
f(x, y) = x' A y
for some matrix A. A tensor generalizes this idea to multilinear forms
f(x, y, z, w, q)
where if you play with the basic idea you'll probably realize that if such an operation were to be represented by some kind of "matrix" A then that matrix would have to be multidimensional. This kind of multidimensional array is a "tensor" although the theory gets a little more complex than just that.
Why isn't any old multidimensional array a tensor? Because we want to see them as arising not as just an arbitrary collection of numbers but instead as representatives of these kinds of multilinear mappings. In particular, we want to know that tensors are invariant under changes of bases. This is similar to how we like to think of matrices as representing "some" linear mapping, but there might be many changes of bases in the source and target vector spaces which change the actual numbers inside without changing the ultimate mapping itself.
Ultimately, to get the fullest understanding of this we would need to talk about vector dual spaces and note that expressions like f(x, y, z, w, q) could involve two "types" of vector spaces, covariant and contravariant, which have different kinds of transformation-under-basis-change properties. To truly be a tensor your multi-dimensional matrix must understand and respect all of these kinds of invariances.
Why isn't any old multidimensional array a tensor? Because we want to see them as arising not as just an arbitrary collection of numbers but instead as representatives of these kinds of multilinear mappings. In particular, we want to know that tensors are invariant under changes of bases. This is similar to how we like to think of matrices as representing "some" linear mapping, but there might be many changes of bases in the source and target vector spaces which change the actual numbers inside without changing the ultimate mapping itself.
Ultimately, to get the fullest understanding of this we would need to talk about vector dual spaces and note that expressions like f(x, y, z, w, q) could involve two "types" of vector spaces, covariant and contravariant, which have different kinds of transformation-under-basis-change properties. To truly be a tensor your multi-dimensional matrix must understand and respect all of these kinds of invariances.