The tensor product is a way of formulating a new tensor from other tensors. If you are given the tensors A, B, C, D, ... then the tensor product TP is also a tensor and is represented by the relation
TP = A@B@C@D@...
The "@" is being used for the product operator which is a symbol which actually looks like an x surrounded by a zero. Suppose A, B, C, D are vectors. We feed in the 1-forms m, n, o, p as follows
TP(m,n,o,p) = A(m)@B(n)@C(o)@D(p)
The value of the tensor TP on the one forms is defined in this way. An example is really trivial and you can call the above an example. The tensors don't need to be vectors on the right. They just need to be tensors. Notice that TP is a tensor of rank 4 since it takes in 4 1-forms.