Well my three favorite apps for root is Titanium Backup (Backs up your apps + data for easy restoring and copies it on SD Card), very flexible application... Otherwise there's Root Explorer which permits you to go through your files easily with a nice little interface and modify system files (BE CAREFUL THOUGH!) and Ad free is another nice one. Blocks almost all known ad servers, so you don't get those pesky ads in applications.
There are a bunch of root apps... Just depends on what you want to do with your phone!
The advantage of root is to gain write access to system folders like /system and the such. Many apps use this to do modifications to the system files.
Also, root will permit you to delve in the All things root forum if you want to try out custom ROMs. Also lets you apply fixes like the Bluetooth pairing not working after upgrading (If you didn't do a factory reset)