Measurement of the thickness or denseness of a fluid. The more dense the fluid the more resistance to flow it exhibits. The "weight" of a motor oil is essentially its viscosity, or how "thick" it is. 30-weight oil is less viscous than 40-weight oil.
In multi-weight oils, such as 10W-40, the first number is the oil's "weight" at low temperatures, and the second number is the oil's "weight" at high temperatures. Multi-weight oils have mostly displaced single-weight oils in the market place because they give better lubrication at start-up, especially in cold weather, than single-weights, and yet give the same protection at higher, operating temperatures.