The Open University course M336 contains two booklets which are dedicated to lattices. One booklet about two-dimensional lattices (GE3) and one about three-dimensional lattices ( and polyhedra ) (GE6). To a layman I would explain lattice as some regular grid of points ( connected by thin lines ).
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In the example above the lattice is defined by two vectors and consists of all points $n \mathbf{a} + m \mathbf{b}$ where $n,m$ are integers.
Fields which use lattice theory are crystallography, finance, game ( maze ) programming, group theory and number theory. When I dug a little bit deeper I discovered that the field of lattices is -ginormous-. Gabriele Nebe and Neil Sloan ( yes him ) maintain a catalog of lattices which now contains over 160,000 lattices. Mathematicians like to generalize over n-dimensions so yes, that database contains lattices in dimensions higher than 3. Like lattices in 40 dimensions for example. Forty.
A catologue of lattices.
Junkyard article about lattices and geometry of numbers.
The mathematical universe is expanding with tremendous speed.
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