I'm a little confused, for the twentieth time, on what tensors are. I thought they were a generalization of matrices-but then they have special transformation rules. I'm looking for a concise definition of what a tensor is. In Griffiths, he says that an $n$ rank tensor transforms with $n$ components of a rotation matrix, why is this? I'll attach what I'm referencing.
Is it a mathematical definition that you truly want? I'm a student myself, and I find the transformation definition to be the most elucidating. It seems the central idea is this: you want the quantities to look a certain way, regardless of the point of view.
This pdf was particularly elucidating, along with Boas' chapter on Tensor Analysis.
Succintly put, all rank-$2$ tensors may be represented as matrices w.r.t. to a particular basis choice. All matrices may be interpreted as rank-$2$ tensors provided you've fixed a basis. The transformation law is then just a consequence of basis independence!