What does a ball of center v and radius r with at most r hops away mean?
An undirected (possibly weighted) graph defines a metric space, in which the distance between two vertices $x$ and $y$ is the length of the shortest path between $x$ and $y$; here the length of the path is the sum of the weights on the edges (weights are assumed to be non-negative), or the number of edges if the graph is unweighted. Given a metric, a ball is defined according to your formula.

If the maximal degree is $d$, the number of vertices in a ball of radius $r$ is at most $1 + d(1+\cdots+(d-1)^{r-1})$, which is what you get for a $d$-regular graph.