The definition of a moon or satellite is a rather fuzzy one, so let us first look at what a co-orbital body is. Loosely speaking, co-orbital means that the two bodies orbit the Sun in roughly the same time period, and are gravitationally linked. Even here, though, the term is used for three distinct purposes.
First, there are asteroids such as 3753 Cruithne which is gravitationally bound to the Earth despite not being in the same orbit, or even the same orbital plane. It goes around the Sun roughly once a year (give or take a few hours), but describes a rather elaborate dance when viewed from the Earth. If one were to take up a position above the Sun's north pole and turn once a year (so that the Earth always stayed in the same apparent position), Cruithne would describe a kidney-shaped "orbit".
Over time, though, the location of the "kidney" with respect to the Earth changes. It moves slowly around the Sun, coming close (well, 15 million kilometers) to the Earth from one side then moving around the sun to approach from the other side, and then back, over roughly 770 years.
At various points over the year, Cruithne is on all sides of the Earth, and so in a very loose sense can be considered to "orbit" the earth. But the distances involved are typically vast, compared to the size of the Earth's orbit, and it is only in a very loose sense that one would consider Cruithne to be a satellite; the average layperson would hardly think of it in these terms at all.
Much more recently, 2002 AA29 was discovered. While Cruithne's dance with the Earth is far more complex than 2002 AA29's relationship, 2002 AA29 has a closer relationship. For starters, 2002 AA29 is (roughly) in the Earth's orbital plane. Cruithne's orbit is inclined more than 15 degrees, while 2002 AA29 has no appreciable inclination.
Instead, 2002 AA29's orbit can be more or less described as a corkscrew around the Earth's orbit. Like Cruithne, it starts close to the Earth and gradually makes its way around the Earth's orbit to approach from the other side some 95 years later.
However, it occasionally comes even closer and gets stuck in a pseudo-orbit around the Earth with a period of one year, for as long as fifty years at a stretch. The next time this will happen will be in roughly 2600 AD. Here, too, the orbit maintains its corkscrew appearance, but now the Earth is in the center. Each loop of the corkscrew takes one year.
2002 AA29 also comes much closer to the Earth than Cruithne ever does. On January 8, 2003, it will be only 12 times as far away as the Moon; that's only 4.6 million kilometers away.
A third sort of co-orbital motion has been known for a long time. Here the asteroid follows the same orbit as the planet (or other larger body) but never comes close, such as the Trojan asteroids of Jupiter. There are two places on Jupiter's orbit, 60 degrees ahead of and 60 degrees behind the planet where groups of asteroids have been discovered. These two locations are known as the L4 and L5 points, where L stands for Lagrange. (The other three Lagrange points do not lie on the orbit of the planet, and indeed are unstable, so a co-orbital body found there will not be co-orbital for very long.)
Scientists started naming the asteroids in the forward group after Greek heroes of the Trojan war, and the rear group was named after the Trojans. The name "Trojan asteroids" soon referred to both groups, and indeed to other instances of asteroids found in Lagrange points, such as the six Martian Trojan asteroids known to lie in the L4 and L5 points on the orbit of Mars.
Clearly, though, Trojan asteroids do not meet the definition of satellite in any sense of the term. They are gravitationally bound to the planet, and are interesting to study in their own right, but are conclusively not satellites.
So, having looked at three different examples of co-orbital motions, have we reached any definite conclusions as to what a satellite is? Well, no. There does not seem to be a hard and fast rule as to what a satellite is. Clearly the Moon is a satellite and a Trojan asteroid is not, but what about Cruithne or 2002 AA29? The best solution is perhaps one favored by some scientists, who have started to favor the term "quasi-satellite" to describe these elaborate (yet relatively stable) co-orbital motions. While somewhat awkward and thus unlikely to gather any momentum in the popular press, this term does concisely describe the somewhat fuzzy line between satellite and Trojan, and is likely the best one can do, given the wide range of orbital motions one can find.