Let $G$ and $H$ be graphs. The graph $G\times H$, called the *tensor*, *direct* or *categorical product* of $G$ and $H$, has vertices $V(G\times H)=V(G)\times V(H)$, and has an edge between $(u, v)$ and $(u', v')$ whenever $u$ is adjacent to $u'$ in $G$ and $v$ is adjacent to $v'$ in $H$.

I believe you're describing the product $K_2\times K_n$. If we label each vertex by a pair $(i, j)$ where vertices in the same "parallel row" are assigned the same $i$, and vertices "in front of" each other are assigned the same $j$, then your description says that two vertices $(i, j)$ and $(i', j')$ are adjacent if and only if $i\not=i'$ or $j\not=j'$. Since two vertices in a complete graph are adjacent if and only if they are distinct, this is just the same adjacency conditions as in the definition above.

The categorical product is an incredibly useful (and well-studied) concept in the study of graph homomorphisms. Chapter 2 of *Graphs and Homomorphisms* collects a number of results relating to this product. There are many other product operations defined for graphs, which are useful in other contexts; I believe the book *Product Graphs: Structure and Recognition* has more on them.