The term double planet has several accepted usages. What they all share in common is a useful categorization of two planetary bodies of comparable mass that is profoundly involved in the interaction of those bodies over time.
In the most common usage, a double planet is a set of two planets of comparable mass orbiting one another.
The second usage for the term double planet, in the context of the theory of the origin of the Moon (i.e. Earth's Moon), is a set of two planets of comparable mass that collide with each other - i.e. with at least transiently overlapping orbits. A double planet in this sense occurred in the very early Solar System, consisting of the proto-Earth and a second, Mars-sized planet that collided with it at an oblique angle, in the consensus theory of the formation of the Earth-Moon system. The second body was not a proto-Moon because most of its mass was incorporated into the Earth, while the Moon formed from a small fraction of debris kicked up from the Earth by the collision. These double planet precursor bodies to the Earth-Moon system had roughly comparable mass - i.e. a mass ratio in the neighborhood of 10:1. This happens to be similar to the mass ratio of Pluto-Charon.
A third usage has arisen since 1995 when we began to discover extrasolar planets in other Solar Systems. In this context, the term double planet system is used to refer to another Solar System in which two planets have been discovered orbiting the star. As of 2003, there were ten known star systems outside our own with at least two detected planets, qualifying at least as double planet systems (or otherwise as multiple planet systems with more than two planets, which we might expect to find as our observations grow more precise).
In the first usage of "double planet", there has been some debate on precisely where to draw the line between a double planet and a planet-moon system. In most cases, it's simply not an issue because the moon is of very small mass relative to its host planet. In particular, the Earth-Moon and Pluto-Charon systems are the only examples in our present Solar System where the mass of a moon is larger than one fortieth of one percent of the mass of the host planet (i.e. mass ratio of 0.00025 or less). On the other hand, the Earth and the Moon have a mass ratio of 0.01230, and Pluto and its moon Charon have a mass ratio of 0.147. A commonly accepted cutoff point is when the center of mass that the two objects orbit around (the barycenter) is not located under the surface of either body, in which case the barycenter is in space between the two bodies. This literally makes the difference between whether one body orbits around the other body, or whether both bodies orbit about a point in space between them. By this definition, Pluto and Charon qualify as a double planet and Earth and Moon do not. (The issue of whether Pluto should be defined as a planet at all, or is instead simply the largest of the Kuiper belt objects, is a separate matter).
An alternative definition of double planet that was advocated by Isaac Asimov is whether the two bodies in question orbit the primary star in convex orbit s, in which case, the Earth-Moon system counts as a double-planet as the Moon primarily orbits the Sun, with minor peturbations in its solar orbit caused by the Earth. However, this definition has not received attention in the professional literature, perhaps because it is biased to include ever smaller moons at ever larger orbital radii, contradicting the basic qualification of comparable mass. However, the term "double planet" is an informal one, without an official definition. Hypothetically, the moon/planet mass ratios between the two bodies could vary anywhere between some significant margin above zero through one, where the former case represents whatever cutoff ratio is selected to distinguish from a planet-moon system, and the latter case represents two planets of exactly the same mass orbiting each other.
-  - Clyde Tombaugh (1906-97) Astronomer who discovered the Solar System's ninth planet, Nature 385 (1997) 778 (Pluto and Charon are "the only known example of a true double planet".)
-  - It's not easy to make the Moon, Nature 389 (1997) 327 (comparing double planet theory of Moon formation and Pluto-Charon as double planet)
-  - Geochemical implications of the formation of the Moon by a single giant impact, Nature 338 (1989) 29
-  - Occurrence and Stability of Apsidal Resonance in Multiple Planetary Systems, Astrophysical Journal 598 (2003) 1290