I think for most things it's more productive to look at the Laplacian of the graph $G$, which is closely related to the adjacency matrix. Here you can use it to relate the second eigenvalue to a "local vs global" property of the graph.
For simplicity, let's suppose that $G$ is $d$-regular. Then the normalized Laplacian of $G$ is $L= I - \frac1d A$, where $I$ is the $n\times n$ identity, and $A$ is the adjacency matrix. The nice thing about the Laplacian is that, writing vectors as functions $f: V\to \mathbb{R}$ like @dkaeae, and using $\langle \cdot, \cdot \rangle$ for the usual inner product, we have this very nice expression for the quadratic form given by $L$:
$$
\langle f, Lf\rangle = \frac{1}{d} \sum_{(u,v) \in E}{(f(u) - f(v))^2}.
$$
The largest eigenvalue of $A$ is $d$, and corresponds to the smallest eigenvalue of $L$, which is $0$; the second largest eigenvalue $\lambda_2$ of $A$ corresponds to the second smallest eigenvalue of $L$, which is $1 - \frac{\lambda_2}{d}$. By the min-max principle, we have
$$
1 - \frac{\lambda_2}{d}=\min\left\{\frac{\langle f, Lf\rangle}{\langle f, f\rangle}:\sum_{v \in V}{f(v)} = 0, f \neq 0\right\}.
$$
Notice that $\langle f, Lf\rangle$ does not change when we shift $f$ by the same constant for every vertex. So, equivalently, you can define, for any $f:V \to \mathbb{R}$, the "centered" function $f_0$ by $f_0(u) = f(u) - \frac{1}{n}\sum_{v \in V}{f(v)}$, and write
$$
1 - \frac{\lambda_2}{d}=\min\left\{\frac{\langle f, Lf\rangle}{\langle f_0, f_0\rangle}: f \text{ not constant}\right\}.
$$
Now a bit of calculation shows that $\langle f_0, f_0\rangle = \frac{1}{n}\sum_{\{u,v\}\in {V\choose 2}}{(f(u) - f(v))^2}$, and substituting above and dividing numerator and denominator by $\frac{n}{2}$, we have
$$
1 - \frac{\lambda_2}{d}=\min\left\{\frac{\frac{2}{nd} \sum_{(u,v) \in E}{(f(u) - f(v))^2}}{\frac{2}{n^2}\sum_{\{u,v\}\in {V\choose 2}}{(f(u) - f(v))^2}}: f \text{ not constant}\right\}.
$$
What this means is that, if we place every vertex $u$ of $G$ on the real line at the point $f(u)$, then the average distance between two independent random vertices in the graph (the denominator) is at most $\frac{d}{d - \lambda_2}$ times the average distance between the endpoints of a random edge in the graph (the numerator). So in this sense, a large spectral gap means that what happens across a random edge of $G$ (local behavior) is a good predictor for what happens across a random uncorrelated pair of vertices (global behavior).