You have a switch at the bottom of the stairs and a switch at the top of the stairs, and both will turn the stairs light on and off. How does that work?
There are two ingredients to this. First, we need an extra wire running up
the stairs, and second, we need a slightly more complicated kind of switch,
called the "double throw" switch. Here's a cartoon of this kind of switch:
. When the switch is "up" it connects the wire on the
left to the upper wire on the right, while when it is "down" it connects to
the lower wire. Here is the complete diagram for the stairway light: