The two previous writeups describe the history of the humbucker, but neither of them explain how it works.
As mcSey says, a humbucker is a pickup (or for that matter, a microphone) with a second coil inside it. Conventional microphones are essentially loudspeakers in reverse - a coil with a magnet inside it that's connected to a diaphragm (the speaker cone). When current changes in the coil, the magnet moves in and out, moving the cone, and making noise. In reverse, if noise moves the cone, the magnet moves, and current is induced in the coil.
However, as well as current being induced by the moving magnet, general electromagnetic interference in the air can also induce current in the coil. This is most common when a microphone cable is run nearby a mains cable, giving a 50 hertz (or 60 in the USA hum.
Adding a humbucker removes this. The second coil sits behind the first coil, and is exactly the same size. However, it has two differences.
- It's connected up backwards - this means that any induced current is induced with the opposite polarity.
- It has no magnet inside it connected to a cone - this means that the only current induced in it is from interference, and none from sounds in the air.
So, we now have two signals. One is "positive" sound and "positive" interference, and the other is just "negative" interference. Add the two signals together - and hey presto, we have interference-free sound!