The Zodiac sign for the month of Ellul is Virgo (a young girl), which is reflected also in the verse,
'Return, young girl of Israel, return to these towns of yours!' (Jeremiah 31:20),
This concept is behind the designation of the month of Ellul (the last month of the year before Rosh Hashanah - Judgement Day for mankind) as the time of Teshuvah - Repentance. The Prophet Jeremiah uses this parallel between a 'young girl' and Israel on several occasions, as does Isaiah; the Prophets use of the term 'return' implies 'returning to G-d', or repentance.
When the Children of Israel sinned over the Golden Calf, Moses went back up Mount Sinai and begged for G-d's mercy and forgiveness. At the end of 40 days, G-d was appeased and told Moses to,
'Hew out two stone tablets, like the first ones' (Exodus 34:1, Deuteronomy 10:1).
On the 10th of Tishrei (Yom Kippur), Moses descended the mountain with the second set of Commandments on this pair of Tablets.
The 40 days (30 days of Ellul and the first 10 of Tishrei) have therefore been designated as days of repentance, so that sins can be forgiven. Selihot, additional prayers, are said to ask for forgiveness. The Sephardi custom is to say these throughout the entire month; others begin on the fifteenth of the month. The Ashkenazi custom is to say them first on the Sunday (or Motzei Shabbat - Saturday night) immediately preceding Rosh Hashanah, if the latter falls on Thursday or Shabbat, but from the previous Sunday (or Motzei Shabbat), if Rosh Hashanah falls on Monday or Tuesday, so that they are always said for at least four days.
At the end of the morning and evening prayers (morning and afternoon according to the nusah sefarad) during the month of Ellul, Psalm 27 is said. In the morning, this is preceded by blowing the Shofar in memory of the Shofar that was blown when Moses ascended the mountain to obtain the second set of Tablets (the Commandments). The blowing is also to warn the people of the approach of the Day of Judgement.
Ellul in Hebrew can also be understood as an acronym of the first four Hebrew words of Song of Songs 6:3,
'I belong to my beloved and my beloved is mine',
where 'my beloved' is interpreted allegorically as a reference to G-d, and 'I' as referring to Israel. Each of these four Hebrew words ends in the letter yod, which has the numerical value of 10: using gematria, four tens make 40, and this is said to refer to the 40 days of Teshuvah - repentance, from the first of Ellul through the tenth of Tishrei.