$K_{3,3}$ is a complete bipartite graph with $6$ nodes split into $2$ groups of $3$ nodes. All of nodes in one group are connected to all of the nodes in the other groups, but not with nodes in the same group. Here's what it looks like:
However, what is a graph of $K_{m,n}$ supposed to look like? From my understanding the graph will have $m+n$ many nodes, $m$ nodes with a degree of $n$ and $n-m$ nodes with degree $n$. For example, $K_{2,3}$ looks like:
Is my understanding correct? If not, please explain how $K_{m,n}$ is supposed to look like.
Here is respectively what $K_{1,3}$, $K_{3,5}$ and $K_{4,8}$ look like.
In each graph, the two sets of the bipartition are on the left and on the right respectively.
Your $K_{2,3}$ graph is correct but not your $K_{3,3}$ since it misses $3$ edges.