The most common transducer type in headphones is the dynamic driver. It's basically a miniature version of the large driver you'd see in a typical speaker in the home. There are other headphone driver types, but for this article we'll focus on the dynamic driver.
The dynamic driver is a transducer that converts an electrical signal into an acoustic signal. The image above shows a partially disassembled headphone capsule with the earpad and driver grill removed. The shiny circular transparent object is the driver diaphragmit's like the speaker cone in a regular speaker. The electrical signal to the headphones causes it to move in and out (as show at right) and creates the sound you hear in the headphones.
To understand exactly how wire, electrical signal, and magnets cause the diaphragm to move, we'll need to start with electromagnetic principles...and I'll have to break out the Dremel tool and distroy some perfectly good headphones. RIP V-Moda Crossfade LP, you died for a good cause.