From AMS Glossary
An array of functions that obeys certain laws of transformation.
The motivation for the use of tensors in some branches of physics is that they are invariants, not depending on the particular coordinate system employed. The tensors (e.g., stress tensor) that enter into meteorology are of a particularly simple kind, called Cartesian tensors, so the important theorems of tensor analysis play no role, but notational convenience justifies their use. A one-row or one-column tensor array is a vector.