The Islamic calendar (or Hijri calendar) is a purely lunar calendar. It contains 12 months that are based on the motion of the moon, and because 12 synodic months is only 12 x 29.53=354.36 days, the Islamic calendar is consistently shorter than a tropical year, and therefore it shifts with respect to the Christian calendar.
The calendar is based on the Qur'an (Sura IX, 36-37) and its proper observance is a sacred duty for Muslims.
The Islamic calendar is the official calendar in countries around the Gulf, especially Saudi Arabia. But other Muslim countries use the Gregorian calendar for civil purposes and only turn to the Islamic calendar for religious purposes.
The names of the 12 months that comprise the Islamic year are:
Muharram
Safar
Rabi' al-awwal (Rabi' I)
Rabi' al-thani (Rabi' II)
Jumada al-awwal (Jumada I)
Jumada al-thani (Jumada II)
Rajab
Sha'ban
Ramadan
Shawwal
Dhu al-Qi'dah
Dhu al-Hijjah
Important Islamic dates
Names of the days of the week
Islamic Months
GENERAL ARTICLES
The Islamic Calendar
The Sacred Months - IslamWeb
The Four Sacred Months - Tafseer Ibn Kathir - AbdurRahman.org
SEE ALSO
Conversion of Islamic and Christian dates
Islamic calendar - Wikipedia
The Umm al-Qura Calendar of Saudi Arabia