1. It's a body which is in orbit around the Sun and not a satellite of a planet. e.g. the Moon is ruled out as being a planet because it orbits the Earth.
2. Its shape is close to being the shape if it was made of water - e.g. close to a sphere. If the object is spinning then centrifugal force will cause the radius at the equator to be larger that the radius at the poles - so it's never a perfect sphere.
3. It has a clear area around it. In other words it has sucked up or pushed away any other objects to become the dominant body in the area. It has been argued that Neptune should be a dwarf planet for the reason that objects like Pluto keep wandering through its orbit space. However Pluto has been locked into a phase in which it orbits the sun twice for every three orbits of Neptune. If this were not the case Pluto's orbit would be unstable. Therefore Neptune's much larger gravitational force is in control.
When is a satellite a satellite?
You may wonder how to define when one object is a satellite of another. When two objects are locked together by gravity then both objects can be said to be circling each other. The point about which they circle is called the barycentre. If one object is heavier than the other then the barycentre will be closer to that object. When one object is much much heavier then the barycentre can lie inside that object - which is an unofficial definition of a satellite. E.g. in the case of the Earth and the Moon, they orbit each other around a point about 1700km under the surface of the Earth, hence the Moon is a satellite of the Earth.
It is known that Pluto and its moon Charon orbit about a barycentre outside of both objects. They are in fact a binary system - but currently the classification remains that Charon is a satellite of Pluto.
So if it's not a planet... what is it?
What's an SSSB?
If an object fulfils only the first criteria - e.g. it orbits the Sun and is not a satellite of a planet - is not spherical and doesn't dominate its orbit then it's a "Small Solar System Body" (SSSB).
What's a Dwarf Planet?
If an object fulfils the first and second criteria - e.g. it orbits the Sun and is spherical, but doesn't dominate the space around it then it's a "Dwarf Planet". This 1 minute video explains....
So... Which objects are planets, and which are dwarf planets?
The International Astronomical Union (IAU) recognises 5 dwarf planets: Ceres, Pluto, Haumea, Makemake, Eris. However, it is estimated that there are hundreds to thousands of dwarf planets in the Solar System, mostly hiding the Kuiper Belt (a large region of space just outside Neptune’s Orbit). Some astronomers have put forward over 300 objects that are dwarf planet candidates. However the International Astronomical Union (IAU) are reluctant to classify these objects as planets until it can be proved that they meet all the requirements, and due to the problems of studying these small and remote objects it may take some time before reclassification occurs.
In 2008 it was decided to have a sub classification of Plutoids which means dwarf planets which orbit outside (mostly) of Neptune. There are 4 Plutoids : Pluto, Haumea, Makemake and Eris.
And SSSBs? Well there are loads of them, and they include all the asteroids and comets.
Names of all the Planets
Click here to see the names of all the planets and their satellites.