It is like a vector, but it has more components. Common tensors are the Inertia tensor, stress tensor, and strain tensor. Tensors have an algebra and a calculus. In three dimensions a tensor will have nine components. They have property called rank. Scalars are tensors of rank 0, vectors are tensors of rank 1, and the tensor of rank 2 is our inertia, stress, strain, or other quantity.
The number of components that a tensor will have in three dimensions depends on the rank of the tensor. In general, it will be N^R where N is the number of dimensions and R is the rank of the tensor.
You might find this handout from OhioStateUni helpful https://www.physics.ohio-state.edu/~ntg/263/handouts/tensor_intro.pdf Edit Thank you for the vido, spook I will add that Fleisch skates over two difficulties. Firstly the difference between maths and physics that leads to vectors being simple low order tensors and tensors being simple (uncomplicated) vectors. Secondly he mentions the tie up between arrays (matrices) and tensors. Not all arrays are tensors, but all tensors can be written as an array. .
More on tensors and electromagnetic fields. http://en.wikipedia.org/wiki/Electromagnetic_four-potential Being the modern representation of the "physical" EM system. http://en.wikipedia.org/wiki/Electromagnetic_tensor http://en.wikipedia.org/wiki/Aharonov–Bohm_effect
Things like this, repeated time and time again, always amaze me and provide strong evidence, to me, that "math is real" and not just a human creation. We develop a set of mathematical models for a system that describes the system as we know it today, and then study the math behind the model and predict subtle behaviors that were previously completely unknown and, at times, seemingly impossible. Then, often decades later, those behaviors are demonstrated experimentally.
I read an article years ago titled "mathematical reality" that explored the subject so deeply that it even discussed its theological implications. Unfortunately, the link has been removed and I can't seem to find it anywhere.
It amazes me too and also makes my brain hurt trying to 'grok' it. The 'mathematical models' are not just made up to describe physical existence in the present, the correct models of fundamental details predict future (sometimes strange) behaviors of the physical system as conditions change. In this case the physical are things we can't directly 'see' but they exists in the same physical sense as a mountain. So I agree, the 'math' is already there and we are discovering it not inventing it.