This is a very short summary of the key thoughts of Epicurus' thinking. I think most of these flow from a few core beliefs:
A. There are no gods we should fear -- there is no afterlife; celestial and atmospheric phenomena are not the expression of diving wrath; there is no standard of justice imposed upon us by the gods.
B. Pleasure is the highest good, and pain the greatest evil.
From these, everything else follows. Of course, given the Greek mythology, I find it a little hard to fault him on #1, though in the end he's wrong. As far as I can tell, Judaism was unknown to the Greeks at this time. Socrates, Aristotle, and others had both concluded that humans had both spirit and body, but Epicurus seems to conclude that we are just body.
From believe #A he infers a pretty twisted definition of justice -- "There never was an absolute justice, but only an agreement made in reciprocal association...providing against the infliction or suffering of harm." (#33) In other words, there is no absolute justice, just what society agrees to construct. This same philosophy is very present today, standing in stark contrast to the claims of the Declaration of Independence, and addressed at length in Mere Christianity. He ends up with a very utilitarian, relative, and perennially shifting view of justice ("for the time being it was just", #37), where the utility relates to the removal of fear and the embrace of pleasure. How sad.
I find one of his doctrines very apropos to statements I've heard from modern philosophy: "If you fight against all your sensations, you will have no standard to which to refer, and thus no means of judging even those judgments which you pronounce false." (#23) One point for Epicurus on this one.