When you root your device, as I mentioned previously, you flash a special Recovery software to it. Usually it is Clockworkmod. The recovery has an option of creating an image of Boot, System, Data, Cache (various sections of the internal memory responsible for loading the software). This is called a Nandroid backup. It's a mirror image in the sense that once you create it at a point, whenever you restore your device with that image, the device resets to this point of time. All data, including your settings on your device (including wallpaper, downloaded files, saved bookmarks, Missed Calls etc) are restored to that point in the past. It's extremely useful if you like playing with various ROMS, or have a tendency to play around with the system, potentially bidding for havoc.
A Nandroid Backup is your lifesaver.
However, a Titanium backup is an application which saves data, apps etc, but not the boot, and certain parts of the system. It does save Contacts, Missed Calls, Apps, App data etc, but it is not a mirror image. You are not restoring to a point of time, but rather you are fusing old data with your new data. For certain system files, it may have the effect of adding old data to new data, or old apps to newer apps. For others, it may have the effect of corrupting the data too (in rare cases).
Well, a Titanium Backup is useful if you want to change your current ROM. After installing the new ROM, if you dont want to manually install all the apps you were using in the old ROM, you use Ti to restore the apps. If you want to restore the old ROM and apps as it was before you flashed the new ROM, as if you never even flashed it in the first place, you use the Nandroid Restore. The batch restore of Ti is extremely useful, saves a lot of clicks, and helps you play with apps.
Whenever you do a system intensive task like flashing a new ROM, do create both Nandroid and Ti backups. You never know when you need them.