1. Your ethernet connects to your broadband, and your broadband connects to the Internet.
2. Your broadband might be a DSL modem, it might have a cable modem, it might be "something else".
3. Whatever it is, you probably ran some kind of Windows "setup" when you first installed it.
You now need to do the equivalent for Linux.
4. Suggestion: contact your broadband provider, and ask them if they have any instructions for Linux.
That's the easiest way - and, heck - you might get lucky!
5. Plan B:
When you go into the "network administration" GUI, and you see your Realtek, specify "DHCP" (or "dynamic address" - the exact wording will depend on your GUI). Again - you might get lucky.
5. Failing all else, please specify:
a) Your Linux distro and version
b) Whether your broadband is cable or DSL
c) Any other pertinent details (such as your DSL modem's model#, for example)
Here are a couple of links that might also help: