TITLE: Geometrical Interpretation of Tensors (intuition)
QUESTION [9 upvotes]: How would one describe to a non-mathematician (an undergraduate physicist actually- so please do use mathematics-i am not even sure if they can be described without mathematics!) what do tensors represent geometrically?

REPLY [2 votes]: We can add something to the already excellent answer given by Ben Grossmann.
The wonderful Gravitation by Misner,Thorpe and Wheeler (affectionately called MTW) on page 75 describes the most general (m,n) tensor  as:" a linear machine with n input slots for n 1-forms and m input slots for m vectors, given the requested input, it puts out a real number..."
(1-forms are simply the linear operators that take a vector and give out a number).
Tensors are useful in physics because (roughly):
each observer is described by a different 4D basis in spacetime and
each measurement is described by the value of a specific component of a specific tensor in the observer's basis.
Tensor components in one basis are (linearly) related to the components in another basis, so if we know what one observer is measuring, we can find out what another observer will measure.
In short: measurements made by different observers are related in the same way that certain tensor components are related when passing from one basis to another basis