To be classified as a planet, an object must be large enough to be spherical – that is shaped like a ball – and it must be much larger than all the other objects near it. So For that reason Earth is a planet because it is ball-shaped and it is much bigger than the Moon, the only object very close to it. $
The reason Pluto is not longer classed as a planet is that although it is ball shaped, it has a moon called Charon that is only a little but smaller than it. So on these grounds it is not a planet. If you included it you would also have to include other object found far out in the Solar system as planets too, such as Sedna (see http://en.wikipedia.org/wiki/90377_Sedna if you want to read about Sedna)
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