The tensor product $V \otimes W$ of two Vector spaces $V$ and $W$ is a vector space with basis given by $v_i \otimes w_j$, where $v_i$ and $w_j$ are a basis of $V$ and $W$, respectively.

We can also define an operation $\otimes: V\times W \to V \otimes W$, which takes $v \in V$ and $w\in W$ to $v \otimes w \in V \otimes W$, and which is bilinear.

In terms of Category theory, it can be defined as being the vector space $V\otimes W$ which has a universal property that if $B: V \times W \to U$ is a bilinear map to a vector space $U$ then there is a unique linear map $\beta: V \otimes W \to U$ such that $B(v,w) = \beta(v \otimes w)$.

We can also define $V \otimes W$ as the dual space of the space of bilinear forms on $V \times W$.

See here.

We can extend the product to products of many vector spaces. We write $\otimes^p V = V \otimes ... \otimes V$ ($p$ times). With the Direct sum of these we can construct the Tensor algebra.

Tensor and wedge products are Functorial under Linear maps

Can also extend tensor product to Vector bundles