Fainter and more distant celestial bodies, which can't be seen with a naked eye, generally got their titles from the people who found them. In keeping with historic trends, those scientists typically opted for names of ancient Greek and Roman gods. The English astronomer William Herschel supposedly wanted to name Uranus “Georgian Sidus” after King George III but was unsuccessful. Pluto was christened by 11-year-old Venetia Burney, a schoolgirl who suggested the name to her well-connected grandfather, who then got it approved by researchers at the Arizona observatory where the (now dwarf) planet was discovered.
We take these names for granted today, because English has become the international language of science. Most scientific journals are published in English. The International Astronomical Union (IAU), the organization responsible for assigning designations to celestial bodies, is based in France but does its business in English. When it was established, the IAU pretty much adopted all of the English designations for objects in our solar system.
These days the IAU has rules for naming new celestial bodies, which allow for a bit more creativity than the parameters of the major planetary designations. After all, there's a limited number of Greek and Roman gods, and we're finding more and more stuff in space every day.
Broadly, planetary nomenclature reflects the identity of the planet in question: features on Venus (named for the Roman goddess of love) are all named after women; features on the Martian moon Deimos (which is itself named for the Greek god of terror) get their designations from authors who wrote about Mars. Some of the naming schemes are whimsical: craters on the asteroid Gaspra are named after spas of the world. Others are nerdy: clusters of hills or knobs on the Saturnian moon Titan are named after residents of Middle Earth.
These days, researchers on a specific mission — say, one of the Mars rovers — will compose lists of possible names that they can pull from as they discover new mountains, craters, ridges, etc. These informal names are used for initial exploration and research, then submitted to the IAU for ultimate approval. For classes of features that don't have an IAU naming scheme, scientists are free to indulge their wackiest impulses. When scientists on the Spirit Mars rover mission had to come up with classification system for soil types, they used flavors of ice cream; now the Martian landscape is littered with rocks called “Cookies and Cream” and “Mudpie." (It's worth noting that the IAU does not approve those kinds of names — they're just for informal use).
If you are hoping to get something in space — perhaps a nice asteroid, or a crater on a minor moon — named after yourself, you are out of luck. It's extremely uncommon to name celestial bodies after people, and that honor is typically only given to prominent scientists who have passed away.
Likewise, no matter what anyone tries to sell you, there is no way to “buy” the name of a star. This stuff is a scam! You can pay as much as you want to sites like StarRegistry.com, but all you're going to get in return is a piece of cardstock with some fancy lettering on it. The name you picked won't be recognized by anyone outside the company that took your money. The IAU makes this point on its website in no uncertain terms.
“As an international scientific organization, the IAU dissociates itself entirely from the commercial practice of 'selling' fictitious star names or 'real estate' on other planets or moons in the Solar System,” it says.
Stars in named constellations, which are visible to the naked eye, are typically given a letter of the Greek alphabet according to their position in the constellation. For example, the brightest star in Cygnus (the Swan) is Alpha Cygni, the next brightest is Beta Cygni, etc. Some of these stars also have names of their own (Alpha Cygni is also known as Deneb).
That may be a disappointment to astronomy lovers who like the idea of a star named for themselves, their spouse, grandparent or favorite TV show character. But as the IAU points out, “like true love and many other of the best things in human life, the beauty of the night sky is not for sale, but is free for all to enjoy.”
And if that does not satisfy you: Purchasing names “would generate a system of mounting confusion for no factual reason,” the IAU says. “And this is the opposite of what taxpayers pay scientists to do.”