Before you try Hibernate on a laptop, make sure you know how to pop the battery out to force a reboot. It's really common for laptops to have problems with coming back from Hibernate, but it's not so universal that you shouldn't try it.

I second wende's comment. The "hibernate" function on my laptop ought to be labeled "coma".

"Standby basically turns off power consuming components like the hard disks and monitor. It switches the computer to a low power state. Its much like a warm boot. Any contents of memory and unsaved desktop settings are lost. Hibernation saves state information by writing a hibernation file which contains the contents of memory and is thus the same size as total RAM. This is a snapshot of active memory. When you turn your PC back on, the state, including which applications are running (desktop) and the memory contents are restored to RAM and voila! - you are back to where you were when Hibernation mode started.
Hibernation is just short of a complete shutdown and takes longer to get things going again.

You should absolutely not be losing settings when you go into Standby mode - Standby simply shuts down components like the hard drive and graphics card, and puts the processor and other system devices into their low-power state. It takes the machine a few seconds to recover when you come back out of Standby. If you're losing settings you're doing something wrong. Keep in mind that on most machines, you don't need to hit the power button to come out of Standby - just touching the keyboard will restore many systems (you just need to wait a moment). If you hit the power button you may actually turn the machine off.

Hibernate saves the current state of the machine to the hard drive and pretty much shuts the whole system off. Coming back from Hibernate can take about a minute, depending on the machine, and some older systems can have trouble with it. Network connections need to be re-established, as do communications with peripherals like scanners or printers. I don't use Hibernate much because there can be stability issues with a machine once it returns from Hibernate, but I think typically you need to hit the power button to get a machine out of Hibernate. Consult your owner's manual.

Windows also allows you to control the power settings of individual components. You can set your hard drive to spin down after "X" number of minutes of disuse, for example. It can save you a lot of power if you're willing to put up with the occasional stall while some component is coming back online.