The term originated from Universal Genève, which in the 1940s named Tri-Compax a watch with three distinct functions: chronograph, full calendar, and moonphase. Today the meaning has narrowed, and quite loosely: almost everywhere "tricompax" simply indicates the arrangement of three chronograph subdials on the dial. Those who use the term in its original sense and those who use it in its current sense don't always understand each other, and it's worth knowing before opening your mouth in a negotiation.
In its modern and widespread version, the tricompax is a three-register chronograph: small seconds, 30-minute totalizer, 12-hour totalizer. The three subdials occupy the symmetric positions at 3, 6, and 9 o'clock, with variations depending on the caliber. The most common configuration in Valjoux calibers of the 1950s and 1960s placed the small seconds at 9 o'clock, the minute totalizer at 3 o'clock, the hour totalizer at 6 o'clock. Some texts cite different arrangements — minutes at 12 o'clock, seconds at 9 o'clock — and this depends on the specific caliber, not on a single rule. I've disassembled enough Valjoux 72 and Valjoux 23 movements on the bench to know that there is no canonical position valid for all.
The difference from the bicompax is structural: the bicompax has only two registers, usually chronograph minutes and chronograph hours, and the small seconds either are missing or are incorporated into the central hand. The rattrapante is another category altogether: it doesn't add a subdial, but doubles the seconds hand to measure intermediate times without stopping the main chronograph.
Visually, the three subdials at the three lower cardinal positions produce what is called in the trade a "smiley face": the axis of the two side subdials forms the eyes, the lower one forms the mouth. It's an instinctive reading, and it explains why this configuration has lasted for decades without being replaced by more modern layouts.
The tricompax is associated with the great manual-winding calibers of the postwar period — Valjoux 72, Heuer 2444 and 3336, Universal Genève — and today it is taken up by both manufactures and builders who use derived ébauches. The date, when present, is often inserted in the window of the hour totalizer or in a separate window, as in the Monaco's Calibre 02, without altering the basic structure.