[chobytes]

My version is just: Tensors allow us to write data and operations on data in a way which does not depend on how we chose to represent them.

For example, if I have a vector x in V and a map T from V to W, then I would like the truth of T(x)=y to be independent of how I represent T and x.

[zardo]

I like the concrete example from when I first used tensors in school. Stress in a block of concrete. You can choose any basis you like to represent the stresses and transform between them.

Whether or not the concrete block breaks under that stress obviously does not depend on your choice of basis or units, so your transformation rules had better reflect that reality.