When a rare or popular DX station comes on the air, hundreds of stations call at once. That crowd is a pile-up. Working one takes a little technique and a lot of restraint, because the DX station is hearing a wall of noise and picks out whatever call sign it can.
Why the DX station splits
If everyone called on the DX station's own frequency, nobody could hear anyone. So the DX transmits on one frequency and listens on another, spreading the callers across a window. It also keeps callers off the DX's own frequency, reduces interference, and avoids asking people to transmit on a frequency that may be outside some callers' privileges.
Illustrative and not to scale. Offsets vary: the DX station announces where it is listening.
You hear only the DX station's side, on frequency f. Your radio listens on f and transmits in the window the DX announced. Many radios do this with two VFOs and a SPLIT switch; the mechanics are in RIT, XIT and split operation.
Blue: where you listen. Red: where you transmit. Offsets are examples.
Working a pile-up
Listen first, for a while. Learn where the DX is listening, how wide the window is, how long each contact takes and how it picks callers (by call sign area, by the last letters, by whoever stands out).
Set up split and put your transmit frequency in the listening window.
Pick a spot in the window. Avoid sitting on top of the station the DX is answering at that moment, and avoid a spot already thick with callers.
Call with your full call sign, once or twice. Not the last two letters, not the DX call sign, not your grid or your name.
Listen after each call. If the DX answers someone else, wait. Calling over a contact in progress helps nobody.
When you hear your call, answer with the minimum: a report and perhaps "thanks". The DX station is working fast.
Do not call again once logged. A duplicate costs the DX station time and others a chance.
Etiquette
Never transmit on the DX frequency (f in the diagram). That is where the DX is trying to talk.
Stay in your privileges. If the DX is listening where you are not allowed to transmit, you cannot call there. Your license limits go with you.
Do not tune up on or near the DX frequency. Use a dummy load or a clear spot.
Do not coach the pile-up. Telling others how to behave on the air only adds noise. The DX operator manages the crowd; your job is to be a good example.
Deliberate interference is prohibited. If someone is jamming, do not answer or argue.
Be patient. If the signal is strong, the DX will eventually get to you; if it is not, move on gracefully.