If a phone is not rooted that means that apps are securely sandboxed. There is no way that one app can access the data held by another app. In the underlying linux filesystem they are actually represented as different users, and do not have permission to access each others' files.
The only things that can backup the files are a) the system itself and b) the app can backup it's own files.
In Froyo (2.2) Google introduced cloud backup, which many app developers have integrated into their apps. This means than some apps will just automatically restore their data as soon as you sign into a new phone with your Google account. However, this will only work for those apps that have implemented this functionality.
You should find that all of the system apps now have full backup, or cloud sync, from Froyo onwards.
From my recent experience upgrading to a Galaxy Nexus - Cloud Sync: GMail, Calendar, Contacts; Cloud Backup: All system settings (I was surprised by some), Email settings. Not backed up: Browser bookmarks (though these are synced in ICS).
There is not default system functionality in Vanilla android that will backup all of your apps, and I'm not aware of any operator/manufacturer customisations which have implemented a system-level backup.
One thing that you can do is to install a new recovery, such as ClockworkMod. This does not require your phone to be 'rooted', however, it does need you to have an unlocked bootloader, which I suspect is what you are trying to avoid, as it is typically reflashing the bootloader where 'bricking' occurs.
You may be able to manually pull all of the files from the /data partition, using adb. You can browse them using 'adb shell', and pull them individually using 'adb pull', but you would need root to be able to restore them. This also requires a certain amount of technical understanding. edit: actually after further testing, even this requires root access.
The suggested MyBackupPro below will not backup all of your apps - only those with exposed contentResolvers (ie SMS, calendar, contacts etc). Those things are automatically backed up in Gingerbread anyway. (I can't comment or vote down)