An N-dimensional matrix that operates on N vectors to yield a number. N is also called the rank. A matrix is 2 dimensional, i.e. N = 2 A vector is 1 dimensional, i.e. N = 1 A scalar is 0 dimensional, i.e. N = 0
The number of subscripts on a tensor indicates the rank. e.g. for Spacetime Metric, both tensors have two subscripts and are therefore tensors of rank 2.