There are multiple levels of standby. The best one is hibernation, or S4, which shuts down everything and saves the OS state to a file on the hard drive; the computer is effectively turned off entirely in this mode, just like if you hit the ATX power switch, and can be unplugged if you want and not lose data. Booting up is much faster than if you'd shut the system down via the OS, because it doesn't have to process all the bootup files and start any applications, they're already loaded directly into memory in their previous state along with any open files. S4 is implemented in the operating system, the BIOS doesn't have anything to do with it really.
S3 is "suspend to RAM", which stores the OS state in the RAM, and shuts down all other devices and components leaves just a trickle of charge to keep the RAM refreshing. This allows an almost instant boot-up, because the all that has to be done is spin up the fans and hard drive, but the OS state doesn't have to be read off the hard drive. All data is lost if you unplug the power or switch off the PSU. The BIOS has to support this. This is most likely the mode that your computer is using when it goes into standby.
S1 is most likely the state you are thinking ought to happen. It leaves pretty much everything running, but shuts off the high-draw devices like hard drives, and the CPU just stops processing instructions. Many mainboards come with S1 as their default power management setting for when the OS instructs to go into standby or sleep mode. It uses much more power than S3 mode, but doesn't really start up much faster since you still have to wait for the hard drive to spin up.
The OS is not aware of whether the system will be using S1 or S3 when it is instructed to go into sleep mode, as that is controlled by the BIOS. (S2 is rarely implemented.)
http://en.wikipedia.org/wiki/Advanced_Configuration_and_Power_Interface