In the end, the IAU decided to create an entirely new class of objects - known as 'dwarf planets' that differ from planets in one key sense. At the moment their definition of a planet means the following -
1. That it orbits the sun
2. That it is large enough to have been rounded by its own gravity (can be anywhere from 200-900 km across, depending on its composition, also known as achieving 'hydrostatic equilibrium')
3. That its gravitational influence has 'cleared the neighborhood' of its orbit of other large objects
A dwarf planet meanwhile, is an object that has only fulfilled the first two criteria.
Given this definition, as of 2014 there are now eight planets and five dwarf planets in the solar system. The planets, which most people should be familiar with - include Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune. The so-far designated dwarf planets are generally less well known, and include Ceres (previously the largest body in the asteroid belt) and Eris as well as the more recently discovered objects Haumea and Makemake. It is expected in the coming years that hundreds if not thousands more roughly Pluto-sized objects will be spotted somewhere beyond the orbit of Neptune.
Pluto itself of course was caught in the middle of this interplanetary crossfire, and was thus downgraded to a dwarf planet as well.
Some astronomers however, have disputed the IAU's definition of a planet, and its not hard to see why. Calling these objects 'dwarf planets' is obviously something of a misnomer, as the third criteria for them to fill above is not just based on their size, but also on the distance at which they orbit the sun.
To explain - 'clearing the neighborhood' means that an object is sufficiently massive that its own gravity is able to disrupt any other large objects from sharing its orbit. This is why, for instance, none of the major planets happen to orbit particularly close to each other. Venus orbits about twice as far out from the sun as Mercury, and the Earth and Mars are separated by gaps roughly as large. Saturn is similarly about twice as far from the sun as Jupiter, and Uranus and Neptune continue this trend among the gas giants. It would be impossible for any of these large planets to share orbits remotely close to one another, as within a relatively short while (geologically speaking at least) their mutual gravitational attraction would grow sufficient either for the two objects to collide, or have a near miss in which one will likely be flung out of the solar system entirely.
Originally, models of planetary formation predict, there should have been hundreds of planets orbiting the sun early in the solar system's history, but within a few million years most would have suffered either of these fates, eventually leaving us with the eight major planets we have today, as well as countless smaller bodies of course.
Now how does this upset our definition of 'dwarf planet'? Because whether a planet has 'cleared its orbit' or not is not determined purely by its size, but by the length of its orbit as well. The required size of a 'planet' will increase proportionate to its distance from the sun.
In other words, if we took a small planet like Mars, and plunked it out somewhere beyond the orbit of Neptune (or more precisely, more than 146 AU away from the sun - one AU being the average distance between the sun and the Earth) then according to the IAU, it would no longer be considered a planet at all. Even the Earth, beyond 3,000-odd AU from the sun, would drop from being a planet to a mere 'dwarf planet' without any actual change in its size.
Conversely, if any of the dwarf planets happened to be closer to the sun, they would automatically be upgraded to planet status despite not having gained an ounce of weight. In Pluto's case for instance, this would be approximately within the orbit of Venus (thus for those wishing to have the IAU reinstate its 'planet' definition, all you'd need to do is attach a rope and tow it to somewhere between the orbits of Venus and Mercury...though simply bribing them might be easier).
Therein, lies the misnomer. It is perfectly possible (one may even say likely) that in the near future we will discover one or more 'dwarf' planets beyond the orbit of Neptune that is actually larger than Mercury or Mars (or possibly even the Earth, though we'd likely have spotted it by now).
There are other problems with current nomenclature regarding astronomical bodies as well, ones that will probably not be resolved until such a borderline case is found that the IAU (or whatever future body may make such decisions) is forced to draw a line somewhere. For instance, what is the definition of a moon? There is no minimum size requirement for a moon (i.e. such an object does not have to be large enough to have been rounded by its own gravity) meaning, for instance, that every individual chunk of rock and ice in Saturn's rings could very well be considered 'moons' in their own right.
Even the dividing line between 'planet' and 'star' is vague. The general definition is based on whether an object is sufficiently large to have undergone nuclear fusion. The problem however is that there are several different types of fusion, each requiring a different amount of pressure to begin. Deuterium burning begins at about 13 Jupiter masses, Lithium burning at 65, and Hydrogen-1 fusion (the 'regular' kind) at 75-80. To deal with this problem, astronomers have created a new category between the two called a 'brown dwarf'. This includes objects of between roughly 13 and 80 Jupiter masses (the sun, by comparison, is just over a thousand Jupiter masses).
As for the issues of defining planets and moons however, here is the classification system I would be compelled to use - and which, at the very least, I intend to use in any science fiction stories I may write, and who knows? Terminology used in science fiction can sometimes permeate into the real. Apparently terms such as 'gas giant', 'terraform', 'droid', 'deep space', 'ion drive' and 'computer virus' were all invented by sci-fi writers -
http://hellogiggles.com/10-words-science-fiction-invented-that-we-use-today
Firstly, instead of 'planet' and 'dwarf planet' I would be compelled to put both kinds of objects under the umbrella definition of 'planet'. Only the first two criteria mentioned above (orbiting a star and large enough to be rounded by its own gravity) would be required.
However, I would then divide the definition of 'planet' into two categories - 'primary' planets and 'secondary' planets. The eight 'proper' planets that we currently have - from Mercury to Neptune, would be re-labelled as 'primary planets' while any dwarf planets so found would become known as 'secondary planets'.
I prefer this definition because it does not directly reference the size of the objects, although at the same time it implies primary planets are larger. It references the fact that the eight 'major' planets are privileged to be not only large, but dominant in their orbits. At the same time it still deems secondary planets as planets, as they may be just as large, but also implies that they are usually not.
Simple yes?
And while I do realize that 'primary' and 'secondary' and kinda big words, we are referring to something of a technical definition. Both kinds of objects could merely be referred to as 'planets' until there was a need to distinguish between them. School children for instance, would simply be taught that instead of there being 'eight planets - and lots and lots of dwarf planets' that there are 'lots and lots of planets - of which there are eight primary ones'. In getting them to memorize the planets, I imagine we'd still stick to having them just name the eight primary ones of course. All we're talking about is a different definition, one that is really no more complicated, and at the same time actually makes more sense.
Now as for moons, there is another divide here we should be making.
A term has come into use lately to refer to particularly tiny moons - generally in the context of the relatively small (and often short-lived) clumps of ice that have been spotted forming regularly in Saturn's rings. These have been referred to as 'moonlets'. The term has also sometimes been used to refer to asteroid moons.
I feel the term is underrated however, and deserving of an upgrade.
Currently there are 180 identified moons of the solar system's planets (and dwarf planets) however, there is a distinction that should be made here. Only a small number of those moons are 'moons' of the sort we imagine when we look up into the night sky at our own - at what is obviously an entire separate world. Most of these 'moons' are little or nothing more than captured asteroids. The two moons of Mars for instance - Phobos and Deimos, have dimensions no greater than 27 and 8 km respectively on their longest axes. This compares to the 3,500 km diameter of our own moon, which is several million times more massive than either of them.
Luckily however, there is a solution to this problem.
We can call upon the old definition that separates planets from smaller objects (what used to be called 'minor planets', but were also reclassified by the IAU in 2006 and are now 'small solar system bodies') that is - that a planet has to have been rounded by its own gravity. Objects like asteroids and comets also orbit the sun, but are declared separate to planets based purely on their size, so why not apply the same standard to objects that happen to be in orbit around a planet?
Under this definition, the nineteen of the solar system's 180 moons that have been sufficiently rounded by their own gravities would remain as - 'moons'. These include our own Moon, IO, Europa, Ganymede and Callisto around Jupiter, Mimas, Enceladus, Tethys, Dione, Rhea, Titan and Iapetus around Saturn, Ariel, Umbriel, Titania, Oberon and Miranda around Uranus, Triton around Neptune and Charon around Pluto.
The other 161 discovered moons would be re-labelled as 'moonlets'. Thus -
As far as our other definitions go, I have fewer qualms. I'm quite happy with the definition of a 'brown dwarf' sitting between planets and stars. Evidently the name is something of a misnomer, as 'brown' dwarfs are predicted to actually have a reddish hue, however 'red dwarf' already refers to low-mass stars with between 75 and roughly 500 Jupiter masses, and would seem even more deserving of that label. Stars are generally categorized by their color, which most closely reflects their mass. Basically, the bigger they are, the fiercer they burn, giving them higher surface temperatures and brighter hues. Stellar evolution is a tricky subject, especially when we start talking about such bizarre forms as white dwarfs, neutron stars and black holes, but the current definitions do sound pretty sensible to me. One problem however, is whether planet-sized objects orbiting around a brown dwarf are indeed 'planets' or merely 'moons'. Personally, I am inclined to think they are the former.
Then at the other end of the spectrum we have objects smaller than planets. These are divided into a number of overlapping categories. 'Asteroid' is a commonly used term, referring to small chunks of rock not large enough to be considered planets. However, the definition sometimes blurs with 'comet' as comets are merely asteroids with significant quantities of lighter elements like water, which sublimates when approaching the sun. The definition wouldn't be too bad, except that many icy asteroids never come close enough to the sun for their surfaces to sublimate at all, making them effectively 'comets-in-waiting'.
Then there is the question of whether there's a lower bound on 'small' solar system objects. Does the definition include meteoroids? That is - objects small enough to burn up in the Earth's atmosphere (generally less than a few meters across, and also begs the question of why the Earth's atmosphere is the benchmark. It is another arbitrary unit, like measuring explosive power in 'Hiroshimas'). What about interplanetary dust as well? Or individual atoms free-floating in space?
To conclude, regardless of these rough edges, the definitions I've outlined above would give us the following solar system - one that consists of one star, eight primary planets and a large number of secondary planets (of which only five have so far been discovered) orbited by nineteen moons and 161 discovered moonlets.
Case closed.