One of OS X Mountain Lion's lesser-known features is the curiously named Power Nap. It's roughly analogous to an actual sleeping mountain lion; while he's catching some Zs in a tree, his brain is constantly aware of what's going on around him, monitoring sounds and scents that might spell danger or food.
Likewise, Mountain Lion's Power Nap will keep your Mac a tiny bit awake. Just enough of the system will be active during sleep (on certain Mac models) to grab email, notes, reminders and messages, run backups, and download OS X updates. All of this is done without turning on a screen, powering up a fan, or doing anything else that could appear to be "waking" your Mac. It's enabled for both plugged-in and battery-only setups by checking the appropriate Power Nap box on System Preferences > Energy Saver.
Power Nap in OS X Mountain Lion works only on current MacBooks that only ship with built-in flash storage -- the MacBook Air (2nd generation, meaning 2011 or later) and Retina MacBook Pro -- so this isn't a feature you're going to see on every Mac today. Will we see this extending to more Macs in the future? I certainly hope so; it could theoretically work on any Mac with proper power management circuitry and a boot SSD.
I've enabled it on my 2011 MacBook Air and it works quite well. Previously, if I hadn't awakened the MBA for several days, I could count on having the Mail app spin for quite a while as it downloaded my email. Now when I open my Air's cover, I can be sure that it's going to be up to date with most, if not all, of my email.