HFC (Cable Modem) MAC authentication:
Most cable companies authorize your cable modem based on it's MAC address. Every DOCSIS cable modem has a unique one. When your modem boots up, it attempts to talk to your cable provider, and the provider will either grant or deny it access based on it's MAC.
HFC (Cable Modem) and NIC (Network Card) MAC authentication:
Some cable companies, for some reason, also authenticate the network card in your computer. First it will authenticate the Cable Modem, and then when your computer attempts to get online, they'll check and make sure that you're still using the same computer/network card/etc that you've always used. If you've changed your network card or installed a router, they will deny you access to the Internet.
This is why most cable routers have a "MAC Address Clone" (Use Computer MAC address) feature. If you tell your router to clone your network card's MAC Address, it makes your cable provider think that you're still using the same computer/network card to access their services, and therefore they allow your router on the network.
For more information, see How can I troubleshoot my cable modem?.