I have an Android tablet that I rooted to attempt to install a different OS. After I failed and restored my tablet to a backup that I had made earlier (it's still rooted), I tried to update Android. It checked for updates, and promptly displayed a box that said "Your device has been modified. Software updates are not available." Is there any way that I can update my tablet while it's rooted?
There are different methods for updating your device. Once you've rooted it, the checksum for your
/system partition changes, and an update might check that and abort because of it (as you just experienced); this is mostly the case with ota-updates.
However: major version updates are rarely spread OTA, but mostly have to be flashed manually using a corresponding computer program. Which one that might be, is manufacturer-specific and depends on your device (so for details on that, we need to know which one it is). Still, applied as an update it might again fail with its checksum verification.
A third variant would be assuming a "virgin device", and make a full install. This usually does not apply any checksum verification (as it does a complete install, and not an update). For this, you might want to check with Where can I find stock or custom ROMs for my Android device?
P.S.: After clarification for the targeted device, there are two more tags you might want to investigate: odin is Samsung's flash-tool for Windows, heimdall is a multi-platform open source equivalent.