Mass is a property of a physical object, measured by its resistance to acceleration when a force is applied. Generally speaking, mass is a measurement of how much matter an object contains. An object's mass also determines the strength of its gravitational pull, which is stronger in objects with more mass (think the Earth vs. the Moon).
Mass is fundamentally different from weight in that weight is actually a measurement of the force of gravity on an object. When we weigh ourselves on earth, we are actually measuring the Earth's gravitational pull on ourselves. Mass, however, is an unchanging property of a given object. What this means is that, while an astronaut will weigh less on the Moon, since the Moon has a weaker gravitational pull than the Earth, the same astronaut will have the same mass, no matter where he travels to.
This is shown by his most famous equation, E=mC2. This is called the mass-energy equivalence equation and means that energy is equal to mass times the speed of light squared. Confused? Don't worry, Einstein was a pretty smart guy.