The complement of the line graph of $K_5$ can be constructed as follows. Label the vertices of $K_5$ as $1,2,\ldots,5$. The 10 edges of this graph are the ${5 \choose 2}$ 2-subsets of $\{1,\ldots,5\}$. The line graph $L(K_5)$ thus has 10 vertices, labeled by these 10 2-subsets $\{i,j\}$. Two vertices $\{i,j\}, \{k,\ell\}$ are adjacent in $L(K_5)$ iff the two 2-subsets have a nontrivial overlap. The complement of $L(K_5)$ is the graph with the same 10 vertices, and with two vertices being adjacent iff the corresponding two 2-subsets are disjoint. This graph is the famous Petersen graph (common drawings of this graph are available online) and happens to arise often as a counterexample to many conjectures.
More generally, the generalized Kneser graph $J(n,k,i)$ is the graph whose vertex set is the set of all $k$-subsets of $\{1,\ldots,n\}$, and with two vertices adjacent iff the corresponding subsets have an intersection of size exactly $i$. Thus, $J(5,2,1)$ is $L(K_5)$, and its complement graph $J(5,2,0)$ is the Petersen graph.