CHANGE VIA: Control Panel > Power Options > Change Plan Settings > Change Advanced Power Settings
Standby puts your computer into energy-saving mode, where it uses very little power.
Hibernate saves your workspace (all your open windows), then turns the computer off.
The difference is that hibernate saves more energy because the computer goes off completely, but it takes longer for the computer to wake up from hibernation, so it's not as convenient.
Sleep initially means Standby, but it switches to Hibernate if the battery level drops too low (laptop) or the computer has been sleeping for more than three hours (both desktops & laptops).You can change the 3-hour period to something else in Settings > Power Options.
In the basic sleep mode, the computer enters a low power-consumption mode, keeping programs and data state in memory; in hybrid sleep mode, the computer enters a low power-consumption mode, keeping programs and data state in memory but also writes the memory content to disk, which means in the event of a power outage the computer's state can be recovered from the disk version of the memory state. Hybrid sleep takes slightly longer to go into low power-mode because the memory content has to be written to disk. On my computer, sleep takes 2 seconds to enter, whereas hybrid sleep takes 16 seconds. The extra 14 seconds is worth it to avoid losing data in the event of a power outage.