All the other planets (Mercury, Venus, Mars, etc.) have names. What's the Earth's name? The other moons in the solar system have names too (Phobos, Io, Titan, etc.). What's our moon's name? Our galaxy has a name: the Milky Way. So what's the name of our solar system?

The name of our planet is the Earth. The name of our moon is the Moon. The name of our solar system is the Solar System.

Notice that I capitalize them, because when used as names, they are proper nouns. This also helps us distinguish between the planet Earth and earth (meaning soil), between the Earth's Moon and moon (meaning the natural satellite of a planet), and between our Solar System and any other solar systems (since any system containing a star and a planet or a planet-forming disk can be called a solar system.)

This is the English language usage approved by the International Astronomical Union, the body in charge of naming celestial objects. It may seem odd that these important objects don't have names, but if you think about it, it just reinforces their importance. For example, the Moon is the Moon, not just any moon. It requires no other name, because it's the most important moon!

You may read or hear people using Luna for the Moon, or Terra or Gaia for the Earth, or Sol for the Sun, but in English-speaking countries, these are poetic terms, often seen in science fiction stories, but not used by astronomers in scientific writing. In some countries where Romance languages are spoken, these terms are the official names.

It's also interesting to note that most astronomers do not call our galaxy the Milky Way in technical writing--they call it the Galaxy.