Rooted android phones are phones that allow the user to control the operating system of the device. This allows users to overcome the limitation placed by service providers and hardware manufacturers. The user of the invention is able to install and run specific applications that require this special privileged. On Apple devices, this is termed jailbreaking.
A rooted device is not qualified for the warranty. Unrooted Android phones are the exact opposite of rooted android phones. The unrooted device remains the way the manufacturers made it. It does not permit the installation of specific applications due to the limitations caused by the wireless service providers and the hardware manufacturer.
You may be hearing that there are some Android phones that are being sold as rooted and there are also some that are not. Rooted Android phones are those that will allow the user to check the ROM menu wherein the ROM is built for the device. This means that rooted Android devices may have more than one OS.
This is very much different from unrooted devices wherein only the original OS placed on the device will be available. The OS of the unrooted device can only be updated when the manufacturer has an update for the OS. For rooted devices, the OS that you may place can be pirated.