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Also known as periodically labeled lattice graphs in graph theory.

Here is a concrete Edge Regular Lattice Graph:

In this graph, the following pattern is repeating locally:

So a Edge Regular Lattice Graph is a Lattice graph $G$, such that in the natural embedding of $G$, each edge label points in the same direction from the perspective of every vertex. Also, the number of edge labels is twice the number of dimensions.

Above we have 4 edge labels in the 2D lattice graph. One for each direction. In a 3D lattice graph, we would have 6 edge labels.