Ok, so basically, power gets to the radio in one of three ways depending on the Radio:
Batteries are used in a couple of different ways:
1st. They are used for direct contact connection such as in a handheld radio. In this application batteries are directly connected through contacts to a circuit board inside of the radio.
2nd. You can connect through cabling to the battery in your car for a mobile radio, for moble radios are designed for a 12volt (car battery) application. Yes there are other applications in which you can connect to a car battery or any other 12 volt battery but we are trying to keep it simple here.
Power supplies are typically used when you want to connect a mobile radio (that is set up to run on a 12volt circuit) to house power. Basically what a power supply does is convert the 120volts in your house down to around 12 volts.
House power is just that: the power you use in your house, typically in the range of 114volts to 126volts (RMS). Typically a radio sold as a "base" will have a plug in the back to just plug in the wall although I would recommend getting a decent surge protector to plug into as a precaution.