A retrograde revolution around the sun, i.e. a retrograde orbit, is not the same thing as apparent retrograde motion to an observer on Earth. While the descriptions of apparent retrograde motion given above are accurate, they do not address an actual retrograde orbit.
All of the planets are orbiting the sun in the same direction as the sun is rotating. If you were to view the solar system from directly "above", it would be fair to compare it to a spinning wheel, moving in a counterclockwise direction. Why don't any of the planets move in a clockwise direction, i.e. a retrograde orbit? There are at least two answers.
1. The formation of the solar system
Based on our current observations, we believe that all of the planets in the solar system were formed from the same cloud of gas and dust that the Sun was formed from. Other, younger stars, as well as the galaxy itself, demonstrate that most of the mass orbiting a larger mass will tend to move in a prograde direction, i.e. the same direction that the central body is rotating in. This does not preclude the possibility of a retrograde orbit occurring, but the retrograde orbit itself is unstable. This is because of tides.
2. Tidal forces
We commonly think of tides as being related only to the ocean, but tides actually affect the shape of the planet as well. The Earth and the moon are literally deforming each other, stretching both bodies into an elliptical shape. This changes their gravitational properties. Tidal deformation occurs between any two bodies which are in orbit with each other, including the sun and the planets.
As the central body rotates on its axis, the bulge produced by tidal deformation moves with it. Since there is more gravitational force concentrated in this bulge, its position relative to the orbiting body will speed that body up, or slow it down. Consider this diagram as though we are looking "down" at the Earth orbiting in a circle:
The arrows represent the direction the earth is orbiting, and the direction the sun is rotating. If the Earth is moving faster than the sun, the massive tidal bulge on the sun tugs it backward, slowing it down. If the Earth is moving slower than the sun, the massive tidal bulge on the sun tugs it forward, speeding it up. Nevertheless it stays in a prograde orbit.
Now consider if this was a retrograde orbit:
Now, the bulge is going to tug AGAINST the planet's retrograde motion, causing it to slow down, and eventually reverse its direction (or, more likely, collide with the Sun).
So how is a retrograde orbit possible at all? It could be the result of capturing a planet that wasn't originally a part of our solar system, or it could be the result of a catastrophic collision of some sort. Either way, we know that a retrograde orbit would not be stable.
We understand that the word “retrograde” means moving backwards, or moving in the opposite direction in consideration to something else. So understanding planet retrogrades is difficult because ideally no planet is supposed to stop or move backwards while making a revolution around the sun. And technically, it isn’t. No planet can, for that matter, move backwards. Planets have fixed paths, fixed speeds.
So, why do sometimes, some planets appear to stop, move backwards and then resume their direction later on? Actually, most of the planets move in an anti-clockwise direction, from west to east. The planets closer to the sun complete their revolutions faster than the ones that are comparatively farther. Retrograde motion is just the outcome of this differential planetary orbital motion around the sun.
Visualize. At some point, the planet that is closer to the sun, for example, will overtake the one comparatively farther. And after reaching a certain place in the revolution circle, it will appear to move in the opposite direction, i.e. from east to west. If we take one of them as primary, usually our Earth, the other seems to be moving backwards. Although, the planet is just moving at the other side of its orbit with different velocity.
You may have observed the effect of differential motion while you travel in your car. The other vehicles, which are moving with lesser speeds, and even the trees, stagnant objects, etc. (with zero speed) seem to flow backwards. But, in reality they are not.
Each planet retrograde has been given a special, deep interpretation in astrology. In fact, Astrologers believe that planet retrogrades have inescapable, generally negative, influences on everyone's life on Earth.
Ptolemy proposed, and it was accepted for over a thousand years, that other planets and the sun all revolved about the earth (geocentric orbits). That required planets to move in retrograde motion. Copernicus proposed, and Galileo proved, that all planets revolved about the sun (heliocentric orbits).
http://csep10.phys.utk.edu/astr161/lect/retrograde/aristotle.html has an animated video that demonstrates retrograde motion.
Looking at the sky, night after night, man no doubt noticed very early that most of the brilliant dots he could see there retained their relative position. A triangle in that corner looked pretty much the same next night, and the night after that; and the same could be said of most of the figures he saw. But there were a few dots that moved against the fixed background of stars. These no doubt provoked great curiosity, and were the subject of very careful scrutiny. The ancient greeks dubbed them wanderers for their relative rambling behavior, and it is from this greek term that their modern name comes: Planets.
The simplest model to explain the motion of the planets has them, along with the Moon and the Sun (the other two wandering objects in the sky), moving in circles, centered around a fixed Earth. This model explains why these celestial objects move against the background of the stars, but there were some oddities in their movements that were not well explained by it.
One of the most notable of these behaviors has to be the retrograde motion of the planets. Planets, as we noted, move relative to the stars, and they do so, generally, moving from west to east in the sky. If we look at the sky, at the same time of the night on two different nights, we will generally notice that the planets have moved a little to the east. But every once in a while, a planet will do something funny: it will slow down its motion, will appear to "stop" for a short while, and will then start moving in the opposite direction, to the west. Eventually, it will stop again, and resume its movement towards the east.
It was unacceptable that a planet would stop in its track and reverse its motion along its circular paths for a short while, to reverse them once again a while later. So, clearly, a better model was needed. Ptolemy devised the most sophisticated of these models that still kept the Earth as the center (so called geocentric models, although to account for other, subtler, irregularities, in his model the Earth was no longer at the exact center). One of its basic ideas was that planets were carried on a small circle (called the epicycle, whose center moves along another, larger, circle around the Earth, called the deferent.
This model worked well as far as retrograde motion was concerned, since when the planet is in the lower part of the epicycle it is moving in the opposite direction as viewed from the earth. But, as mentioned above, there were subtler oddities that needed to be taken into account, too, and Ptolemy was forced to introduce more and more complications into his model to attempt to explain them.