How many moons does Earth have?

The Earth only has one moon, or natural satellite. It is usually called "the Moon" or by its Latin name Luna (from which the adjective lunar is derived).

Other objects identified as "moons" are actually co-orbiting asteroids or quasi-satellites, which follow orbital paths around the Sun, not the Earth. These include the asteroid 3753 Cruitne and 5 other small bodies:
(54509) 2000 PH5,
(85770) 1998 UP1
2002 AA29
2003 YN107

(see the related link below)

Other speculative moons were reported by astronomers in the 19th century:

Lilith, one of the Waltemath's moons which were suggested in 1898 by Georg Waltemath of Hamburg. Supposedly these small moons were difficult to observe because they were "dark" and could rarely be seen. One other astronomer claimed to have seen one in 1918, and named it Lilith. But they were likely optical illusions.
Petit's moon was a fallacious report in 1846 by Frederic Petit of Toulouse. The data from his observations indicated that the perigee of the moon would have only been about 11.7 kilometers (37,000 feet) above Earth - about where airliners fly.
Earth has only one moon.