When it's a proper noun: big "E", no "the" When it's a common noun: small "e", with "the"

It depends. There is technically a difference between ¨Earth¨ and ¨an earth,¨ because ¨earth" without being capitalized is a life-giving planet. ¨Earth¨ is this specific planet. Rule of thumb: if it's ¨the earth,¨ then it's not capitalized.

The only time you need to capitalise this word is when you are naming the planets : Venus, Mars, Jupiter, Mercury, Saturn.... then 'Earth' would need a capital E. At all other times, earth is just a word, not a name, so you don't need a capital letter.