The black is power and the other two are called travelers that wire to the the. On the other end the two white wires are wire nutted together.
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What is the black screw on a light switch. Its used in conjunction with another 3 way switch in another area and they both control a light or set of lights. Or this could be all done from one switch having just the 3 wire running to the other switch. It serves as the second hot wire in the switch loop. At the second switch box location the wiring is similar to the first switch with the traveler terminals connected to the traveler wires coming from the first switch. The black hot wire goes to the brass screw or into the hole in the back of the device on the same side as the brass screw. The black screw would be one for power and the other switch for power out to the light.
A three way switch is different. This type of switch has two brass screws usually on the bottom and a copper or black screw on the top left. This wire is sometimes red. This type of switch lets you turn the light off or on from more than one place in the room. You should also see a green wire called a ground wire which. The wire to the light is connected to t read more.
The black and red wires from the white cable running between the switches get connected to either of the two brass screws on the switch. The green or bare copper ground wire if the device has one attaches to the green screw terminal on the switch or to the electrical box. The black wire from the box is connected to the black screw on the right side at the bottom by the outlet. Ideally the old switch is still there so you can wire the new one up the same way as the old one. These are called terminal screws. You should see two black wires each connecting to a different screw on the right side of the switch.
So lets say the wires you have in box one are all of them. However at this second switch the common screw terminal is connected to a black hot wire that leads onward to the light fixture. The black wire from the circuit breaker panel gets attached to the black screw on the 3 way switch. The black wire from the new cable connects to the black hot wire in the fixture box and to one of the terminals on the single pole switch. The white wire from the new cable connects to the fixtures hot wire terminal or hot wire lead and to the other screw terminal on the switch.