Intuition behind eigenvalues of an adjacency matrix

Question

I am currently working to understand the use of the Cheeger bound and of Cheeger's inequality, and their use for spectral partitioning, conductance, expansion, etc, but I still struggle to have a start of an intuition regarding the second eigenvalue of the adjacency matrix.
Usually, in graph theory, most of the concepts we come across of are quite simple to intuit, but in this case, I can't even come up with what kind of graphs would have a second eigenvalue being very low, or very high.
I've been reading similar questions asked here and there on the SE network, but they usually refer to eigenvalues in different fields (multivariate analysis, Euclidian distance matrices, correlation matrices ...).
But nothing about spectral partitioning and graph theory.

Can someone try and share his intuition/experience of this second eigenvalue in the case of graphs and adjacency matrices?