I'm no nuclear physicist, but here's what I've gathered from various confusingly technical websites:

Assume that all matter is energy. Everything we deal with on a daily basis has a positive energy. Protons, elections, neutrons, photos, radio waves, all of it.

Dirac's equations (based on Einstein's, i believe) predicted that there should also be negative energy. Today we know this energy as antimatter. But how can you have negative energy? Less than none?

Dirac explained this by saying that all measurements are taken relative to empty space, in which all of the negative energy levels have been filled.

Imagine a graph with positive energy levels at the top and negative energy at the bottom. All energy levels below zero are filled in. This is the "Sea of Dirac" or "Dirac Sea".

According to Dirac's theory (as I understand it), antimatter would not be matter with negative mass, but an absence of "negative mass". Antiparticles are the holes in the negative energy sea.