  We introduce the notion of the second lattice width of a lattice polygon and use this to classify lattice triangles by their width and second width. This is equivalent to classifying lattice triangles contained in an $n \times m$ rectangle (and no smaller) 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 $\mathbb{P}(1,1,1,2,2,2)$. 
Classification of lattice triangles by their two smallest widths