Abstract achievements: his Fields Medal winning work on algebraic $K$-theory, plus inventing model categories, homotopical algebra, and an axiomatic approach to abstract homotopy theory.
Applied achievements: his 1964 PhD thesis was about partial differential equations, and was used in many subsequent applied papers on systems of PDEs, including an Annals paper by Goldschmidt. Later, in the 1980s, he did work in Riemannian geometry and functional analysis and he invented the notion of 'superconnection' in differential geometry and analysis. This work has been applied to work on Heat kernels and Dirac operators, Deformation quantization, elliptic operators, index theory, and topological quantum field theory.