The physical meaning of the tensor product

I have come across tensor products many times in physics, namely for matrices, vector-space elements, Hilbert-space elements (quantum states), and representations of groups and algebras. However, the abstract meaning of the tensor product still alludes me. Although I can understand its context in physics, I cannot see that its physical description and its abstract definitions, for example those given on Wikipedia, are equivalent. Could someone provide me with a geometrical, as well as an algebraic, interpretation of the tensor product and also explain why it is required?

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What do you mean, why they are required? From what you tell us, you have seen ample evidence that tensor products are useful... The «abstract meaning» of things is peanuts. Peanuts. – Mariano Suárez-Alvarez Feb 21 at 4:13

The concept of tensor products of vector spaces is required for the description of General Relativity. For example, the metric tensor $\mathbf{g}$ of a pseudo-Riemannian manifold $(M,\mathbf{g})$ can be viewed as a ‘disjoint collection’ of tensor products. In terms of a local-coordinate system $(x^{1},\ldots,x^{n})$, we can express $\mathbf{g}$ as $g_{ij} \cdot d{x^{i}} \otimes d{x^{j}}$, where $d{x^{i}}$ and $d{x^{j}}$ are differential $1$-forms and $g_{ij}$ is just a scalar coefficient. At each $p \in M$, what $\mathbf{g}$ does is to take two tangent vectors at $p$ as input and produce a scalar in $\mathbb{R}$ as output, all in a linear fashion. In other words, at each point $p$, we can view $\mathbf{g} = g_{ij} \cdot d{x^{i}} \otimes d{x^{j}}$ as a bilinear mapping from ${T_{p}}(M) \times {T_{p}}(M)$ to $\mathbb{R}$, where ${T_{p}}(M)$ denotes the tangent space at $p$.
The metric tensor is the very object that encodes geometrical information about the manifold $M$. All other useful quantities are constructed from it, such as the Riemann curvature tensor and the Ricci curvature tensor (of course, we still need something called an ‘affine connection’ in order to define these tensors). In General Relativity, the metric tensor encodes the curvature of spacetime, which can and does affect the performance of high-precision instrumentation such as the Global Positioning System (GPS).
For vector spaces over a field $\mathbb{K}$, tensor products are usually defined in terms of multilinear mappings. You may have seen the more abstract definition using a system of generators, but it can be shown that these two definitions are the same. From the categorical point of view, both constructions satisfy the same universal property (this property is explicated in the Wikipedia article on tensor products), so they must be isomorphic in the category of $\mathbb{K}$-vector spaces.