Earth:
It is lowercase unless at the beginning of a sentence. That would be according to Webster's. Moon is also lowercase, as is sun, universe, world. For proper names we usually resort to Latin and use Terra, Luna, Sol, etc., and then the names are considered proper names and capitalized. It may be used with or without the definite article, as in earth, or the earth. It is sometimes capitalized anyway, depending on whether your English teacher insists.
I very much doubt an English teacher would insist that you use lower case for Jupiter, Saturn, or Mars. Likewise Earth.
When we say "the earth", we are really saying "the planet named Earth."
It is a minor point, though, and my proofreading correction really didn't belong on this physics thread. (But as we all know, there is no slack on FR.)