"Hibernate" saves everything in memory to the hard drive, then shuts down the computer completely. "Standby", on the other hand, is basically a deep sleep mode, where all the processes are suspended and the computer is powered down, but not completely, as is the case with hibernation.
Standby mode is quicker to resume normal operation, but hibernation doesn't drain the battery as quickly as standby does.
However, this will be irrelevant when Vista is released, because M$ is creating a new power-down state that will combine elements of both hibernation and standby, and get rid of those as seperate entities.