Eventually, in 2006, the IAU settled on a three-part definition: a planet orbits the sun, is sufficiently massive to form itself into a round shape by gravity and has “cleared the neighborhood around its orbit,” which means it’s the most gravitationally dominant body there. (This term did initially cause some confusion because it could be misinterpreted to mean that the planet can sweep its orbit completely clean of any and all other bodies, which is impossible.) If a body fulfills the first two conditions but not the third, it’s called a “dwarf” planet.
IAU’s definitions of “planet” and “dwarf planet”
Later: “Objects need to be at least 400 km or so in size to round out via self-gravity”