Previously, the vector space $\mathbb{V}$ and three vector product operations were considered in Simple tensor algebra I and Simple tensor algebra II. But we do not talk much about the tensor. Here, the tensor will be introduced.
So, what is a tensor? The following figure shows a tensor in a 2D Cartesian coordinate system.
The vector is a tensor, and it is a first-order (1st-order) tensor. Actually, the scalar is a tensor as well, we can consider the scalar as a zeroth-order (0th-order) tensor.
Let $ a, b$ be two scalars in $ \mathbb{R}$, we put these two scalars (0th-order tensor) together with $ \{ a, b \}$ to form a vector (1st-order tensor). So, if we can put two vectors $ \boldsymbol g, \boldsymbol f $ together, then we will get a 2nd-order tensor. The tensor product $ \otimes $ plays such role here ($ \boldsymbol g \otimes \boldsymbol f $).
This is not a tutorial, please refer to the books for tensor algebra.