The 12 Months of the Jewish Year starting from Rosh Hashana, Jewish New year are:

 Tishrei

 Cheshvan

Kisleiv

 Tevet

Shvat

Adar

Nissan

Iyar

Sivan

Tamuz

Av

Elul.

Rosh Hashana, or the Jewish New Year is in Tishrei, the first 2 days of the month.

The month preceding is Elul.

In this month there is increased introspection in preparation for the Day of Judgement at the begining of the next month, another term used for Rosh Hashana.

Yom Hadin, Hebrew for Day of Judgement.

The Torah was first given in the first week of Sivan.

Moses went up for 40 days to receive the Ten Commandments from G-d.

But the worshipping of the Golden Calf angered G-d and Moses broke the 2 Tablets containing the 10 commandments. Moses then prayed for 40 days to G-d to have mercy on the people and frogive them and give them the Torah inscribed in the 2 Tablets again.After this second 40 days the Jews were forgiven and Moses went up for a final 40 days to receive the 2 tablets with the 10 commandments inscribed on them again.

This last 40 day period started at the begining of the month of Elul.

Moses came down at the end of the last 40 day period on the day of Yom Kippur, the 10th day of the month of Tishrei.

The month of Elul is looked upon as an opprtune time for repentence and preparation for Rosh Hashana, the day of Judegement when the King of Kings is in judgement over his creations. The month of Elul is viewed as a period of divine Good Will as that is the start of the period after Moses prayed for forgiveness of the people and to receieve the 10 commandments again.