I encountered the symbol "$\otimes$" in a Numberphile video about the Dehn Invariant, and I have no clue what it means.
What, in layman terms, does this symbol do?
I'm not all that experienced in maths and often fail to understand the jargon used in the descriptions of what the tensor symbol does and what a "tensor product" is.
In mathematics vectors are abstract entities that like the vectors used in physics to represent forces can be summed or multiplied by a scalar (i.e. a number).
A vector space is roughly speaking a set of vectors that contains every sum of any two vectors in it and every multiple of every vector in it.
If we have two vector spaces $V$ and $W$ we can build up a new vector space $V\times W$ considering pairs of vectors: an element in $V\times W$ will be a pair $$ (v,w)\qquad v\in V,\quad w\in W. $$ The tensor product $V\otimes W$ is a different and a bit subtler way to pair vectors in $V$ with vectors in $W$. A basic element in $V\otimes W$ is something denoted $$ v\otimes w. $$ The main difference between pairs and tensors is that while the pairs $$ (\lambda v,w)\qquad (v,\lambda w) $$ give different elements in $V\times W$ there is an actual equality $$ (\lambda v)\otimes w=v\otimes(\lambda w) $$ as elements in $V\otimes W$ (in the above $\lambda$ is any scalar).
The basic reason to introduce tensor products is to reduce, so to speak, the theory of multilinear functions to the theory of linear functions. For instance, there's a characterization of the determinant of a square matrix that uses the language of tensor products.