This article is from the Calendars FAQ, by Claus Tondering firstname.lastname@example.org with numerous contributions by others.
The names of the 12 months that comprise the Islamic year are:
3. Rabi' al-awwal (Rabi' I)
4. Rabi' al-thani (Rabi' II)
5. Jumada al-awwal (Jumada I)
6. Jumada al-thani (Jumada II)
11. Dhu al-Qi'dah
12. Dhu al-Hijjah
(Due to different transliterations of the Arabic alphabet, other
spellings of the months are possible.)
Each month starts when the lunar crescent is first seen (by an actual
human being) after a new moon.
Although new moons may be calculated quite precisely, the actual
visibility of the crescent is much more difficult to predict. It
depends on factors such as weather, the optical properties of the
atmosphere, and the location of the observer. It is therefore very
difficult to give accurate information in advance about when a new
month will start.
Furthermore, some Muslims depend on a local sighting of the moon,
whereas others depend on a sighting by authorities somewhere in the
Muslim world. Both are valid Islamic practices, but they may lead to
different starting days for the months.