Lipton's most influential results
The Planar Separator Theorem states that in any planar $n$-vertex graph $G$ there exists a set of $O(\sqrt{n})$ vertices whose removal leaves the graph disconnected into at least two roughly balanced components. Moreover, such a set can be found in linear time. This (tight) result, proved by Lipton and Tarjan (improving on a previous result by Ungar) is a powerful tool for designing algorithms on planar graphs. It gives many exact subexponential time algorithms for NP-hard problems and improved polynomial time approximation algorithms. Looking at the wikipedia page gives a good starting place to explore the numerous applications. An early survey with details of a number of applications was written by Lipton and Tarjan in 1980.