Titanium Backup is one of the most widely used apps for backup/restore apps. This app does a lot of other things like freeze, cache/dalvik cache cleaner and lots of other. TitaniumBackup requires root.
Helium, (in the past the name was Carbon) is the app letting you to backup/restore apps and AppSync to cloud. The whole app is noob-friendly and it's really easy to use. Helium does NOT require root.
EasyBackup is a relatively recently published app with the same purpose as the others - backup&restore apps and also working with cloud services.
This app only backups the .APK files, not the data, therefore root access is not required.
Besides this, it can do a lot of other usefull minor things - shows storage usage, shares market link, quick uninstall feature and other.
The legendary ROM Toolbox, which contains just everyhing you may need, also contains app backup/restore. Requires root access to work.