The standard unit used to measure viscosity is the centistoke (cSt). According to the Automotive and Industrial Lubricants Glossary of Terms:
Viscosity is ordinarily expressed in terms of the time required for a standard quantity of the fluid at a certain temperature to flow through a standard orifice. The higher the value, the more viscous the fluid. Since viscosity varies inversely with temperature, its value is meaningless unless accompanied by the temperature at which it is determined. With petroleum oils, viscosity is now commonly reported in centistokes (cSt), measured at either 40°C or 100 °C (ASTM Method D445 - Kinematic Viscosity).
The centistoke rating is converted into the SAE weight designation using a chart like the one shown on the Superior Lubricants Web site.
Multi-weight oils (such as 10W-30) are a new invention made possible by adding polymers to oil. The polymers allow the oil to have different weights at different temperatures. The first number indicates the viscosity of the oil at a cold temperature, while the second number indicates the viscosity at operating temperature. This page from the Sci.Electronics.Repair FAQ offers the following very interesting description of how the polymers work:
At cold temperatures, the polymers are coiled up and allow the oil to flow as their low numbers indicate. As the oil warms up, the polymers begin to unwind into long chains that prevent the oil from thinning as much as it normally would. The result is that at 100 degrees C, the oil has thinned only as much as the higher viscosity number indicates. Another way of looking at multi-vis oils is to think of a 20W-50 as a 20 weight oil that will not thin more than a 50 weight would when hot.