What is exactly the relation between vectors, matrices, and tensors? I am trying to understand what Tensors are (I have no physics or pure math background, and am starting with machine learning). 
In an introduction to Tensors it is said that tensors are a generalization of scalars, vectors and matrices:
Scalars are 0-order tensors, vectors are 1-order tensors, and matrices are 2-order tensors. n-order tensors are simply an n-dimensional array of numbers. 
However, this does not say anything about the algebraic or geometric properties of tensors, and it seems from reading around the internet that tensors are algebraically defined, rather than defined as an array of numbers. 
Similarly, vectors are defined as elements of a set that satisfy the vector space axioms. I understand this definition of a vector space. And I understand that matrices are representations of linear transformations of vectors.
Question: I am trying to understand what a Tensor is more intuitively, and what the algebraic and intuitive/geometric relation is between tensors on the one hand, and vectors/matrices on the other. (taking into account that matrices are representations of linear transformations of vectors)

Vectors and Tensors are mathematical objects invariant under coordinate system chosen. 
Matrices and Arrays for that matter are representations of rank-2 tensor or vectors in a given coordinate system.
As for the intuitive understanding part, take the analogy of a vector. By definition a vector is a magnitude and a direction. If I said  "5 miles an hour that a way" and point, that's a velocity vector. Until you put a coordinate system on it you can't write it down as an ordered pair. A tensor is the linear transformation that is invariant. Once you impose a coordinate system you can write it down as a matrix. That's why tensors are often used to describe material properties.
