  We introduce the notion of the second lattice width of a lattice polytope and
use this to classify lattice triangles by their width and second width. This is
equivalent to classifying lattice triangles contained in a given rectangle (and
no smaller rectangle) up to affine equivalence. Using this classification we
investigate the automorphism groups and Ehrhart theory of lattice triangles. We
also show that the sequence counting lattice triangles contained in dilations
of the unit square has generating function equal to the Hilbert series of a
degree 8 hypersurface in $\mathbb{P}(1,1,1,2,2,2)$.
