As discussed at Lagrange Points and Trojan Asteroids, there are certain orbital positions where asteroids can be "captured" by planets, in a gravitational tug of war with the Sun. The bodies found near these points are orbiting the Sun, but the way they do so is subtly altered by the gravity of the planets which share their orbits, so that instead of having completely independent orbits of their own, the asteroids follow nearly the same orbital path around the Sun as the planet that controls their orbit.
By far the most noticeable and numerous of these asteroids are the ones which orbit the Sun 60 degrees ahead of and behind Jupiter, almost all of which are named for real (and perhaps in some cases mythological) characters from the Trojan War. That is the reason they are called Trojan asteroids. However, there are a number of other cases in which objects have similar positions ahead of or behind large moons (in which case they are properly called Trojan moons), or other planets (in which case they are called Trojan asteroids, which can be confusing, since the term originally applied only to Jupiter's Trojans). In fact, it was recently discovered that the Earth has a "Trojan" asteroid, orbiting the Sun 60 degrees ahead of us.
The Earth's Trojan Asteroid(s)
It has long been suspected that the Earth might have Trojan-type asteroids orbiting the Sun 60 degrees ahead of or behind us, but detecting them is more difficult than might be expected. From the Earth, such objects would be seen 60 degrees to the east of the Sun in the evening sky, or 60 degrees to the west of the Sun in the pre-dawn sky. That's a reasonably large angle, and in fact more than a dozen degrees further from the Sun than Venus ever gets, so it would seem that it ought to be easy to see objects in such a position. However, there is a fair amount of dust lying in the plane of the Solar System (and therefore, also in the plane of our orbit), which reflects a fair amount of sunlight, causing the area which lies along the Ecliptic to exhibit a glowing arc, referred to as the Zodiacal light. The glow is brightest near the Sun, and fades considerably as you look further from the Sun, so by the time you look 60 degrees away from the Sun, it is too faint to easily notice with the unaided eye. But it is still there, and in telescopic images meant to detect very faint objects the best part of 50 million miles from the Earth (the distance of our Trojan Lagrange points), is brighter than most such asteroids would be. As a result, until very recently, it was not possible to tell whether any small bodies were actually located near the Earth's Trojan Lagrange points.
This uncertainty ended in July 2011, when it was announced that an Earth Trojan had finally been found. The object involved was actually discovered in 2010, but it took some time to study its motion and tell that it really was more or less firmly bound to the Earth's Trojan point. The asteroid involved is about 1000 feet across, which is probably on the large size for objects trapped in that region by the Earth's gravitational interaction with the Sun; but the main thing that helped determine its existence is the fact that its orbit is "unusual" (or at least stated to be so, in the press release announcing its existence). But as discussed below, its motion isn't really unusual, just not the same as the motion of every other (presumed) Earth Trojan.
As Trojan asteroids move around the Sun, they have two motions: (1) an orbital motion identical to the planet which controls their motion, but 60 degrees ahead of or behind the planet, and (2) a back-and-forth, somewhat chaotic motion relative to the Lagrange point which defines that group of Trojans. That point represents a "gravitational well", like a valley in the gravitational field of the Sun, which traps the asteroids, and keeps them moving around the Sun in the same orbit as the planet that controls their motion. But in general, they are not located at the "bottom" of the "valley", but move back and forth, up the side of the valley, gradually slowing as they move away from its center, then back down toward the "bottom", gradually speeding up as they move toward its center. The speed they gain as they move toward the bottom then carries them up the other side of the "valley", slowing as they "rise", until they are forced to move back toward the center again.
In the previous paragraph, "up" and "down" mean away from and toward the Lagrange point. They do not mean up and down relative to the orbit of the asteroid or that of the planet, or for that matter, the plane of the Solar System. Asteroids moving back and forth relative to the Lagrange point can move back and forth in any direction -- in and out relative to the direction to the Sun, or forward and back relative to the direction to the planet, or up and down relative to the plane of the Solar System, or any combination of the three. Which direction they move is a more or less random result of the way they were captured into their current orbits, and gradual changes in that motion as a result of (rare) collisions or (not so rare) near-collisions with other Trojans in the same region.
Since there are several different directions that Trojans could move back and forth relative to their Lagrange point, and any combination of two of them (in and out relative to the Sun, and back and forward relative to the planet) will keep them in the plane of the Solar System, while only one direction (up and down relative to the plane of the Solar System) will move them away from the Ecliptic, most Earth Trojans are probably in or very close to the plane of the Ecliptic, where the Zodiacal light can overwhelm the light from small Earth-Trojans. So it isn't surprising that until very recently, no asteroids had been found in the appropriate places for Earth Trojans. However, the asteroid recently found happens to be moving more up and down, relative to the plane of our orbit, than in any other direction; so it can move sideways relative to the Zodiacal light, placing it in slightly darker parts of the sky, and making its discovery considerably easier. It is in that sense that its orbit is "unusual"; it is not really unusual, just different from other possible orbits in a way that makes it easier to notice. There are almost certainly many other bodies trapped near the Earth's Trojan Lagrange points, but they may all be much smaller, or have orbits relative to the Lagrange point which keep them closer to the plane of our orbit, and if so they will be much harder to find.