Electrical – How exactly does a 3-way switch work

I just finished replacing all of the switches and outlets in my apartment, after fighting a long war with a weird 3-way switch. Needless to say, I won, but I'm still trying to grasp the concept of how it works. I know how a single pole switch works, like so:

-->--HOT-WIRE-IN-->[/SWITCH/]-->--HOT-WIRE-OUT-->--

But how does a 3-way switch work? I know that we have 3 wires coming into the switch, 1) black hot in, 2) black hot out, and 3) red runner, but how does the switch actually work? What does it toggle? When I held up an electricity sensor to the wires, it seemed that wire 1 of course was always hot, but wire 2 and 3 were sometimes hot and sometimes not.

So, how does it actually work? What's being toggled and how?

Also, if you're wondering how to wire a 3-way switch, here's how: connect the black hot in wire to the only black screw, then connect the red runner wire to the brass screw on the same side as the black screw, and the hot out goes on the opposite side on the only brass screw. Just a tip 🙂