Probably the biggest new feature of Leopard is its new backup solution, Time Machine. Apple's website says that "Time Machine is the breakthrough automatic backup that's built right into Mac OS X. It keeps an up-to-date copy of everything on your Mac - digital photos, music, movies, TV shows, and documents. Now, if you ever have the need, you can easily go back in time to recover anything."
First of all, it is dead-easy to set up. Plug in your external drive, and your Mac will ask you if you want to use the drive as a backup disk for Time Machine. Confirm the dialog, and System Preferences will pop up, warning you that the disk first needs to be erased before it can be used by Time Machine. Since I hooked up a brand-new LaCie 500GB external disk, I had no worries about that one. After erasing the disk, Time Machine will start with the initial back up, which logically takes quite a while to complete (as it backs up your entire Mac).
Time Machine will make backups every hour, but since the system works incrementally (using UNIX-style hard links to files and directories), it only writes those files to the backup disk that have been altered since the last backup. It will keep hourly backups of the past 24 hours, daily backups for the past month, and weekly backups until you run out of disk space. When you do, Time Machine will alert you that it is going to delete old backups, so you will not be surprised when old backups suddenly start disappearing.
Time Machine's interface is where it gets quite interesting. When browsing a folder, you can press the Time Machine icon on the dock, and the desktop and all other windows will slide out of view, to reveal a view on a galaxy filled with stars, stars that are passing you buy in a science-fiction-y kind of way. In the middle, Finder windows are set up like a set of domino pieces, with the first window being 'now', and the others being earlier incarnations of the folder in question. At the bottom of the screen are 3D controls that allow you to go back in time, restore files or folders, or cancel the operation. The interface is pretty much self explanatory, albeit a tad bit 'too much bling' for me. I do understand Apple's intentions though: backing up is boring, and they needed to make it sassy. And they surely succeeded at that one.
Time Machine has more tricks up its sleeve, though. Apart from the ability to restore individual files and folders in a versioning-kind-of-way in the Finder, you can also use your Time Machine back ups to restore your computer after a serious crash or hard drive failure. Additionally, Time Machine works in other applications than the Finder too, such as iPhoto or Mail.app. A problem with this is that it is not exactly clear which applications support Time Machine, and which do not. If you click the Time Machine button while using an application that does not support it, Time Machine will show you the desktop folder.
A problem I personally foresaw was how Time Machine would deal with laptop computers, Apple's strongest market. Laptops generally do not have external drives attached to them all day long like desktops and workstations do, and that might pose problems. Luckily, Apple's engineers are actually quite the clever blokes, and they fixed this potential problem by allowing Time Machine to store any changes made in an external drive-less state locally, syncing them with the Time Machine disk as soon as it gets connected again.
I am impressed with Time Machine. It is the cleanest, easiest, prettiest, and yes, funniest back up solution I have ever laid my hands upon. My personal back up habits were a tad bit weird: I have multiple machines running in my house, and I simply synchronized my important documents and photos on all of those machines (manually!); if one installation died on me, I could simply re-install, and copy things over from another machine. Tedious, but it worked. Time Machine is a million times easier, and I can surely see myself relying on this method instead of my OCD-inspired manual synchronisation. A definite thumbs up to Apple and Leopard.