Weight is the measure of how heavy an object is. Weight is measured in standard customary units.
The mass of an object is the amount of material it contains. Mass is measured in standard metric units.
For everyday purposes, when you're on the surface of the earth, the difference is not important. But if you measure something on another planet, its mass will be the same as it is on earth—but its weight will be different. (Weight depends on gravity, and gravity is different on other planets! This is why when you're floating in space, you're weightless. You still have mass, though,)