A simple graph $G(V, E)$ is a set of vertices and edges, where each edge $e \in E$ is defined as $e \subset V$ with $|e| = 2$.
A multigraph generalizes a simple graph, allowing for loops and multiple edges. A loop is an edge that is a one-element subset of $V$. As multiple edges are allowed, $E$ is a multiset of one and two element subsets of $V$.
The hypergraph further generalizes the multigraph, allowing for hyperedges $h$ simply to be $h \subset V$ and $h \neq \emptyset$.
Hypergraphs are incredibly useful in computational biology for modeling complex systems. There are also problems in hypergraphs that are computationally intractable, whereas in simple graphs the problem is quite simple. A good example is $2$-colorability. In simple graphs, it is easy to test if a graph is $2$-colorable, or bipartite. In a hypergrpah, we first have to define our coloring problem such that no hyperedge contains only vertices of one color. With this definition, checking if a hypergraph is $2$-colorable is $NP$-Complete. The proof deals with a probabilistic checkable proof. I actually wrote a paper for a class on this a while back, but I'm a tad rusty on the details.
There are still more general structures that generalize the concept of graphs in some sense, like matroids, for example.
Matroids generalize the concept of linear independence from linear algebra. Graph acyclicity shares this property, as does the property of three or more points being collinear on the Fano Plane. I wouldn't say that a Matroid generalizes the concept of a graph, specifically though.