Imre Lakatos was born in Hungary; he spent hard years, as a Jewish man, during and after the second world war. And he fled from Hungary after the revolution against the Russian regime in 1956, and eventually came to England.
Lakatos was influenced by Karl Popper and George Polya (mathematician), and he finished his doctoral work by writing a thesis on the Logic of Mathematical Discovery submitted to Cambridge in 1961.
(Lakatos in 1961; photo by courtesy of LSE
Lakatos website, maintained by Alex Bellamy)
From 1960 he taught at the London School of Economics, and eventually succeeded Popper's position there, but their views on the philosophy of science diverged.
Lakatos is most famous for his "methodology of research programs" which was formed by a confrontation with Kuhn's view on paradigm. He defended, with Popper, rationalism in the philosophy of science, but took in some elements from Kuhn and inductivism.
According to Lakatos, scientists work under some tradition called a "research program", which has some theoretical assumptions called "hard core", and they try to solve specific problems by adding and modifying auxiliary hypotheses and others. Since the "hard core" is immune to falsification by their "methodological decisions", problem-solving is done by modifications and innovations in auxiliary hypotheses and assumptions which are collectively called "protective belt" around the "hard core". This process produces a history of a research program, a succession of modified theories sharing the "hard core". Lakatos thought that we can meaningfully speak of rationality of adopting, modifying, or abandoning and replacing, such a research program.
See a biography in MacTutor History of Mathematics
New Site on Lakatos (LSE)