Everybody knows that the Earth has a moon: Luna* is roughly a quarter of the size of our planet and is second only to the Sun in celestial brightness. In fact, it is so big relative to our planet's size that it almost qualifies in some scientists' books as a double planet. Luna orbits 384,400 km from Earth, has a diameter of 3476 km, and masses 7.3522 kg. The Moon was also called Selene and Artemis by the ancient Greeks.
However, fewer people know that an asteroid discovered in 1986 is locked in a complex but stable orbit around Earth, making it technically a second moon. The asteroid was named Cruithne (pronounced croo-EEN-ya) and has a a 1::1 resonance with Earth. It takes a year to go around the sun. It is co-orbital with the Earth (meaning it shares the Earth's orbit), but more importantly, it co-rotates with the Earth. This gives it what is known as a "horseshoe" orbit; as the Earth moves, the satellite travels around the Earth, then turns and travels back as if it were following the edges of a gigantic circular horseshoe hovering around the planet. Previous to Cruithne's discovery, such orbits were only theoretical. Cruithne was named after the first Celtic tribe to populate the British Isles -- this tribe is more commonly known as the Picts.
In September of 2002, an amateur Arizona astronomer named Bill Yeung discovered an object in a 50-day orbit around our planet that was briefly thought to be a third moon. Soon after giving the object the designation J002E2, astronomers at the Minor Planet Center in Massachusetts discovered it is actually Apollo 12's cast-off 3rd stage booster. The search for a third natural satellite continues.
Many people had theorized, seriously or fancifully, the existence of a second moon long before before Cruithne was discovered.
In 1846, French astronomer Frederic Petit, then-director of the observatory at Toulouse, claimed to have discovered a second moon in an elliptical orbit around the Earth. While the claim was mainly ignored by his peers, writer Jules Verne learned of it and became intrigued by the idea; he mentions Petit's postulated second moon in his 1865 novel From the Earth to the Moon.
After the publication of that novel, amateur astronomers all over the world rushed to be the first to discover a second moon. In 1898, Dr. Georg Waltemath, a Hamburg scientist, really upped the ante. He announced he'd discovered a second moon inside a whole system of tiny moons orbiting the Earth. In 1926, an amateur German astronomer named W. Spill also claimed to have found a second moon. All such claims, of course, were disproven.
Astrologers particularly latched onto the idea of a second moon. In 1918, an astrologer who called himself Sepharial (his real name was Walter Gornold) declared the existence of a second moon and named it Lilith, after Adam's demonic wife before Eve. According to this astrologer, Lilith was invisible most of the time and only became apparent when it crossed the sun. Lilith, though bogus, captured lots of people's imaginations as bright Luna's black twin representing man's darker nature. It became incorporated into some horoscopes as a result and some astrologers may mention it even today.
Several other authors have since postulated the existence of a second moon. For instance, Samuel Delaney's 1975 novel Dhalgren features an Earth that mysteriously acquires a second moon.
Eleanor Cameron did a more memorable moon treatment in the early 1950s. She wrote a series of children's novels (The Wonderful Flight to the Mushroom Planet and Stowaway to the Mushroom Planet) that are about a tiny, habitable second moon in an invisible orbit 50,000 miles from Earth. The "Mushroom Planet" is covered in various types of mushrooms and is populated by little green people. The premise of course has no scientific credibility, but as a child I found these books delightful.
* Yes, as others have helpfully pointed out, scientists call the moon just "The Moon"; however, all things moonlike are referred to as being lunar, so Luna is still being used as a scientific moniker in some regards. I'm employing the Roman name here to distinguish our major moon from other, lesser or imaginary, moons.