The viscosity grade (for example, 5W-30) tells you the oil's thickness, or viscosity. A thin oil has a lower number and flows more easily, while thick oils have a higher number and are more resistant to flow. Water has a very low viscosity -- it is thin and flows easily. Honey has a very high viscosity -- it is thick and gooey.
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 this page.

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 offers the following very interesting description of how the polymers work:
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