I'm looking for a good definition of sparse graphs. Is a sparse graph effectively a big one, with millions/billions of nodes? An example from real world is the Facebook graph. Or can sparse graphs be in small networks as well?
The common definition is that a family of graphs $G = (V, E)$ is sparse if $m \in o(n^2)$, with $n = |V|$ and $m = |E|$. Formally, that requires the family to be infinite, and we only know something in the limit.
The definition does not apply to graphs of a fixed size per se. One would certainly want $m \ll n^2$ in some sense. For instance, if $m \leq 5n$ or $m \approx \log n$ you would probably say that your graph is "sparse".
You can bridge this gap by requiring a bit more on the asymptotic side. Say we call a family of graphs sparse if
$\qquad\displaystyle m \leq f(n) + O(1)$ with $f \in o(n^2)$;
then we may want to call every graph with $m \leq f(n)$ sparse. The implicit assumption is that the error term hidden in $O(1)$ is well-behaved.
Sparsity is a fuzzy notion, and its meaning will depend on the context. It can be argued a graph is not sparse "by itself", but rather it is sparse relative to other graphs. Indeed, there are several ways to define or measure sparseness.
We can say a connected graph is maximally sparse if it is a tree. Intuitively, a tree-like graph is sparse. But what could be a "tree-like graph"? Perhaps the number of cycles is bounded, or by removing a constant number of vertices we get a tree. One very popular definition of sparseness rises from these considerations. Informally, the treewidth of a graph is a measure of how close a graph is to being a tree.
To address your question, a graph of low treewidth can be either small or large. For instance, forests, trees, and outerplanar graphs all have constant treewidth. In contrast, a complete graph on $n$ vertices has treewidth $n-1$. This corresponds to our intuition that complete graphs are not sparse.
Treewidth is just one measure of sparseness. You can dig deeper into the literature and explore other measures as well.