1. Auto-download purchases
Just like the setting in iOS, you can now automatically download purchases made on your other Macs. Just head to System Preferences -> Software Update and check the box for "Automatically download apps purchased on other Macs".
2. Mini Calendars
In iCal, or Calendar as it is now more simply called, you can now see mini month calendars by clicking the Calendars button in the toolbar.
3. Tap to scroll to top
Another queue from iOS, you can now quickly scroll to the top of your inbox in Mail by simply clicking on the "Sort by" bar. Curious to see if and when this trick will work in other parts of OS X.
4. Specify email notifications
In the General settings for Mail you can now specify what emails you are alerted for in Notification Center. You can, for instance, set to only be notified of emails from your set VIPs (another new feature in Mountain Lion). There's something for all you "productivity" geeks.
5. Chess and Game Center
The famous OS X Chess game now is not only compatible with Game Center, but has all sorts of Achievements to unlock as well (Moves -> Achievements to see them all). Go crazy.
6. Tap for Quick Look
In the Finder, you can use a three finger tap on the trackpad to bring up Quick Look. (The spacebar still works for that too.)
7. Customize the Finder's sidebar
You were always able to arrange folders and whatnot within each section of the Finder's sidebar, but you can now rearrange the sections themselves. Want the Devices section on the top? No problem.
8. Disable asking to keep changes
In System Preferences -> General, you can uncheck the box that reads "Ask to keep changes when closing documents" to disable just that.
9. New wallpapers and photo screensavers
10. Rename files from the title bar
Instead of having to locate the file in the Finder to rename it, you can now just click on a filename in the window's title bar to rename it. This same sort of thing works for bookmarklets too. You can also move a file to another location from the title bar by clicking its name and choosing Move.