Relationship between graph expansion and conductance
There are several different definitions for the expansion of a graph, but all of them are spatial. One common definition defined the expansion $h(G)$ of a graph $G$ as $$ h(G) = \min_{|S| \leq |G|/2} \frac{|\partial S|}{|S|}, $$
where $S$ is a set of vertices containing at most half of the vertices, and $\partial S$ is the (edge) boundary of $S$, the number of edges connecting $S$ to $\overline{S}$. For a $d$-regular connected graph, there is a strong connection between $h(G)$ and a spectral parameter, the eigenvalue gap $\lambda(G)$, which is the smallest non-zero eigenvalue of the Laplacian. For a $d$-regular connected graph, Cheeger's inequality states that
$$ \frac{\lambda(G)}{2} \leq h(G) \leq \sqrt{2d\lambda(G)}.  $$
This is often put in terms of the second largest eigenvalue $\lambda_2(G)$ of the adjacency matrix of $G$, which satisfies $\lambda_2(G) = d - \lambda(G)$. There must be versions of Cheeger's inequality for non-regular graphs, and I'll let you look them up.

A lot of information on expander graphs can be found in the survey (based on course notes) Expander graphs and their applications by Hoori, Linial and Wigderson. Other information is scattered among lecture notes and even recent papers.