"Double planet" is an informal term used to describe a planet with a moon that may be large enough to be considered a planet in its own right; one definition requires that the objects orbit a centre of gravity that is above their surfaces. The formal term is "binary system".
The most widely accepted recognition is when the systems centre of mass (barycentre) is outside of the primary planet's surface. With Pluto and Charon the centre of gravity is outside of Pluto's surface.
However Asimov made a very interesting point, and a definitely more future-sighted point. He said that the measure of a double planet is of the tug-of-war between planet, satellite and star. With the Earth-Moon system, the Sun is winning the tug of war, eventually the barycentre will exit the Earth's surface and eventually the Moon will break free of Earth's orbit. This would make the Earth-Moon system a double planet. However Pluto is winning the fight in the Pluto-Charon system, and eventually Charon will impact Pluto's surface.
I personally prefer Asimov's definition. It's only a matter of time (maybe longer than the sun even) until the barycentre leaves the Earth's surface. IIRC ~600 million years the day lengthens by ~2 hours. This means that likely by the end of Earth's habitability (roughly 2 billion years from now) the day will be ~30 hours long at the present rate (it's considerably speeding up).
The vast majority of planet-satellite systems are arranged in a way that the satellite will eventually collide with its host planet. Those that don't will eventually hit their own classification as a planet or dwarf planet along the way, especially the Moon.
