Abstract achievements: his Fields Medal winning work on derived functors, plus a whole new approach to algebraic geometry that has shaped generations of mathematicians after (see EGA, SGA, FGA)
Applied achievements: his PhD thesis was in functional analysis, and his early papers focused on "the theory of nuclear spaces as foundational for Schwartz distributions" (among other topics). This work has applications to stochastic PDEs, elliptic PDEs, probability theory and mathematical statistics, physics, engineering, and kernel functions (hence, data analysis and machine learning). Following citations on Google Scholar is a fun way to see citations to Grothendieck's work from a wide variety of application areas. Grothendieck also published a paper (cited 60 times) on solution spaces to a general class of PDEs.
