After reading all the answers and commenting on some of them, I now feel ready to supply my own answer, which I hope the OP will find gives an intuitive feel for why tensors are so useful and interesting.

Tensors are a mathematical object, a generalization of scalars, vectors and matrices. But they are not just any generalization, they are a generalization carefully chosen to express the late 19th century early 20th century insight into the laws of physics, that in order to faithfully represent a physical quantity, a mathematical object has to transform in a certain way under certain transformation groups, so that the physical quantity itself is invariant. For example, the distance between two points remains the same when you rotate the laboratory by 180 degrees. Similarly, Newton's laws of motion will still be excellent approximations when you rotate the laboratory too. But at the turn of the 20th century, we realized that not only the rotation group, but also the Lorentz group was important, physical laws had to remain unchanged under Lorentz transformations, too. So physical quantities had to remain invariant not just under rotations, but under Lorentz transformations. Finding which quantities remain invariant under the latter is not as easy as the former, so the development of tensors became more important than before.

But before I even mention an explicit transformation law, I would like to point out that armed only with the knowledge above, I can already see that a scalar, being a single number whose "transformation law" is "don't change", is a very simple tensor. A plane vector is only a little more complicated, having two coordinates, and a transformation law that is rotation of vectors: (x', y') =Rot(x,y) where 'Rot' is a 2x2 matrix based on sinθ and cosθ.

A good example of a 3x3 matrix as a tensor is the moment of inertia matrix, which you can read about in Wikipedia: Moment of inertia interesting that they do not even try to express the required transformation law; but the new physical quantity expressed by the invariance of this tensor is the "moment of inertia' itself, which is a physical property of the rotating object.

Now I have decided against trying to state the general rule for what the explicit transformation law must look like, since I think that would get too detailed. You have to, after all, choose a coordinate system, express the transformation law in those coordinates using an appropriately chosen representation of the transformation group...

That would be a bit much when the goal is intuitive insight. For this goal, better to focus on the notions of invariance, covariance and contravariance rather than try to be too general: tensors are invariants with components (in a given coordinate system) that transform either covariantly or contravariantly under transformations given by the chosen transformation group. This is why they are so general, capable of expressing any physical quantity.