Divine Revelation

They are

1. Muharram
2. Safar
3. Rabi' al-awwal (Rabi' I)
4. Rabi' al-akhir (or Rabi' al-thani) (Rabi' II)
5. Jumada al-awwal (Jumada I)
6. Jumada al-akhir (or Jumada al-thani) (Jumaada II)
7. Rajab
8. Sha'aban
9. Ramadan
10. Shawwal
11. Dhu al-Qi'dah
12. Dhu al-Hijjah

Why is the Islamic Calendar so important?

Allah says in the Quran 9:36
Translation: The number of months in the sight of Allah is twelve (in a year) so ordained by Him the day He created the heavens and the earth; of them four are sacred; that is the straight usage. So wrong not yourselves therein and fight the pagans all together as they fight you all together. But know that Allah is with those who restrain themselves.

Explanation: Allah says that the new moon comes up only once each month since He created the universe.Thus the year has always been of twelve months. Of these 12 months 4 months are Haraam or Prohibited i.e. in which no war must be waged and there should be peace.
Those 4 months are:

1. Dhu al-Qi'dah
2. Dhu al-Hijjah
3. Muharram
4. Rajab

However if the mushriks do not desist from fighting then Muslims must fight them and put up a united front.

Allah says in the Quran 9:37Translation: Verily the transposing (of a prohibited month) is an addition to unbelief: the unbelievers are led to wrong thereby: for they make it lawful one year and forbidden another year in order to adjust the number of months forbidden by Allah and make such forbidden ones lawful. The evil of their course seems pleasing to them. But Allah guideth not those who reject faith.

Explanation: The pagan arabs would practice something called nasi in 2 ways:

A)Whenever it suited them they would declare a prohibited month as ordinary and would continue with their fighting robbery and murder.
They would then declare an ordinary month to be prohibited so as to compensate and make up for the lost number of prohibited months.

B) They would add an extra month into the lunar calendar so that it could sync with the solar year and the all important Hajj would thus fall in the same season of the year saving them from a lot of inconviniences.

By making this practice of nasi unlawful Allah gives the believers clear messages:

A) No human being has the right to turn the unlawful (Haraam) into lawful (Halaal). This is a right that is exclusive to Allah.

B) Allah wants Muslims to follow the lunar calendar so that important months like Ramadhan and Dhu al Hijjah fall under different seasons. For example if the month of Ramadhan circulates among all the seasons of the year then Muslims will get trained to obey His Commandments be it summer winter rain or spring. Similarly for Hajj the pilgrims have to travel during all seasons favourable or unfavourable. It is obvious that this training will help them stand in good stead as they struggle to establish the Religion of Truth in this world.

Suggested reading for further detailed study.
Basic information with some good questions
Wikipedia link has exhaustive information