Short answers, pulled from the story.
The International Astronomical Union defined a planet as a celestial body that orbits the Sun, is massive enough for its own gravity to make it round, and has cleared its neighbourhood of smaller objects. This definition was voted on by 424 astronomers at the IAU General Assembly in Prague on the 24th of August 2006, and it placed Pluto in the new category of dwarf planet.
Pluto was reclassified because the discovery of Eris in 2005, a trans-Neptunian object more massive than Pluto, forced the IAU to formalise a definition. Under the 2006 IAU definition, Pluto meets the first two criteria for planethood but fails the third: it has not cleared its neighbourhood. Pluto accounts for only about 7 percent of the material in its orbital zone, and its orbit lies within the Kuiper belt.
Mike Brown and his team at the California Institute of Technology announced the discovery of Eris on the 29th of July 2005. Eris is a trans-Neptunian object confirmed to be more massive than Pluto, and its discovery forced the International Astronomical Union to act on a formal definition of planet.
Greek astronomers called planets asteres planetai, meaning wandering stars, distinguishing them from the fixed stars that held their positions relative to one another. The five bodies they counted as planets were those visible to the naked eye: Venus, Jupiter, Mercury, Mars, and Saturn. Graeco-Roman cosmology often added the Sun and the Moon to make seven planets in total.
The geophysical definition holds that a planet is any body large enough to reach hydrostatic equilibrium, meaning gravity has pulled it into a round, ellipsoidal shape, regardless of where it orbits or what else shares its orbital zone. The IAU definition adds a third criterion requiring the body to have cleared its neighbourhood, which is what disqualifies Pluto. Some planetary scientists, including Alan Stern, prefer the geophysical definition.
Plutoids are a subclass of dwarf planets comprising trans-Neptunian dwarf planets, announced by the IAU executive committee on the 11th of June 2008. The class includes Pluto, Eris, Makemake (named the 11th of July 2008) and Haumea (named the 17th of September 2008), but excludes Ceres, which orbits in the asteroid belt rather than beyond Neptune. Objects qualify if their absolute magnitude is brighter than plus one, under the assumption that they are likely to be dwarf planets.