Eid Al-Banat (or Hag HaBanot or Rosh Hodesh LaBanot in Hebrew and Fete de Filles in French) is a holiday celebrated by women on Rosh Hodesh Tevet (which falls on Hanukkah). No one knows for sure when people started celebrating it. There are accounts of it being celebrated throughout the Mediterranean (specifically Libya, Algeria, Morocco, Greece, and Turkey), but it is best kept and widely still celebrated by the Tunisian Jewish community today. The holiday's essence is women sisterhood and celebration of women.
This holiday is connected with many events in Jewish history:
Why Rosh Hodesh?
רש"י מגילה כב ע"ב
ראשי חדשים - אין בו ביטול מלאכה כל כך שאין הנשים עושות מלאכה בהן והכי נמי אמרינן במס' ראש השנה (ד' כג.) גבי משואות משום ביטול מלאכה לעם שני ימים ושמעתי מפי מורי הזקן ז"ל שניתנה להם מצוה זו בשביל שלא פירקו נזמיהן בעגל (תוספות. ואני מצאתי בפרק מ"ה דברייתא דרבי אליעזר שמעו הנשים ולא רצו ליתן נזמיהן לבעליהן אלא אמרו להן אתם רוצים לעשות פסל ומסכה שאין בו כח להציל ונתן הקב"ה שכרן של נשים בעולם הזה שיהו משמרות ראשי חדשים יותר מן האנשים ולעוה"ב הן עתידות להתחדש כמו ראשי חדשים שנאמר תתחדש כנשר נעורייכי ע"כ) ומקרא מסייעו דכתיב אשר נסתרת שם ביום המעשה (שמואל א כ)...
... That this mitzva was given to them [the women] because they did not give away or take apart their jewelry for [the sin of the] golden calf. (Tosafot. And I found in perek 45 a braita of Rabbi Eliezer [that says that] the women heard and did not want to give their jewelry to their husbands, rather they said to them 'you want to make a statue and idol that does not have power to save' and Hashem gave them a reward of women in this world that they will be the keepers of Rosh Hodeshes more than the men and in Olam Haba they are destined to renew like Rosh Hodeshes as it is written "so that thy youth is renewed like the eagle"...
But not everyone was so pro women celebrating and taking a day off on Rosh Hodesh:
אליהו בשייצי - אדרת אליהו (ספר הלכה קראי) עמ' 67
והנה הנשים העצלות, האלמנות והטפלות יש להן מנהג ואינן עושות מלאכתן בראש חודש, והן חושבות שהוא באיסור, וזה מצד עצלותן וסכלותן, ועוברות בלא תוסיפו
Eliyahu Bshitzi - Aderet Eliyahu pg. 67
And here the lazy women, the widows and the subordinate, they have a tradition and they do not do their work on Rosh Hodesh, and they think it is not permitted, and this is from their laziness and their stupidity, and pass without adding.
Purim: Esther's Coronation
According to another tradition Eid Al-Banat comes from Esther. Because of the Esther's Coronation, many of the traditions on Eid Al-Banat are similar to those of Purim.
One of the traditions was to throw a mishte nashim (a women's banquet). The women would have a se'uda mishte from the morning until nightfall women would gather and meet to bake sweets and cookies, to eat, laugh, sing, and dance.
Esther was surrounded by women and her relationship to them was that of a sisterhood.
רמב"ם - פירוש מגילת אסתר
(טז) גַּם אֲנִי וְנַעֲרֹתַי אָצוּם כֵּן - אף אני ונערותי היהודיות נצטרף לצום של כלל ישראל. שבלי ספק היו שם בנות ישראל שבויות, אשר בעבר הייתה ושתי הארורה בת הארור מעבידה אותן בשבת
Rambam on Megilat Esther
I and my maidens will observe the same fast - My Jewish maidens and I will join the fast of all of Israel. That without a doubt there were Jewish girls in captivity, that in the past the wicked daughter of the wicked Vashti was working them on Shabbat.
Another similarity Eid Al-Banat has to Purim is the tradition of sending cookies from one household to another, similar to mishloah manot.
Some other traditions are to drink wine, and another is to give to children in need, similar to matanot la'evyonim.
Because the holiday falls on Hannukah, there is also a tradition that connects it to Yehudit and to Hanna bat Matityahu.
Although not canonized in the Tanakh, Sefer Yehudit (whose Hebrew version was lost) tells of the story of Yehudit who slayed Holofernes, the general of the Greek army. Tunisian tradition dates this story to Rosh Hodesh Tevet, and marks its importance with Eid Al-Banat celebrations. According to Mizheyna Cohen from Djerba, Eid Al-Banat "is like a remembrance for Yehudit who in her strength saved Jerusalem and Israel." This is why there is also a tradition to eat dairy products.
The general story goes like this: (taken from aish.com)
"A Greek commander, [Holofernes,] led his army to put down a revolt that was beginning in Jerusalem. The Greek forces encamped around the walls of the city and began a protracted siege. Though Jerusalem was a well-fortified city, the relentless siege by a superior army began to exact a great toll on the citizens of the city. A widow named Yehudit left the city and requested an audience with the commander. [She also brought cheeses and wine.] Her plan was to seduce him and then to kill him.
Her plan succeeded. The commander gave a feast in honor of Yehudit and he became quite drunk. That night the commander and Yehudit retired to his private tent where he soon fell into a deep sleep. While he was asleep Yehudit took his sword and decapitated him.
Yehudit then brought the commander’s head back to Jerusalem where it was hung on the city walls for everyone to see. The Jews were inspired by the daring heroism of Yehudit, and the Greek forces retreated.
It was a key turning point in the Jewish revolt against the Greeks."
Hannukah: Hanna bat Matityahu
According to tradition Hanna incited the rebellion against the Greeks, and thus played an integral role in saving Am Israel.
Fun fact: Apparently during the time that this story happened it was very taboo to talk about what was happening to the newlywed women out of fear that the Greeks would do something worse to them, so no one really talked about this story until afterwards. Some think that this story and the story of Yehudit are actually the same story because when Yehudit went to the enemy camp she prayed, "אל אלוהי אבי שמעון. אתה נתת חרב בידו להנקם בבני נכר על אשר חיללו את אחותו הבתולה", "to the God of my father Shimon. You gave a sword in his hand to revenge the foreign sons on that they tainted his sister the virgin" and it doesn't make sense that a widowed woman would mention the story of a virgin. Now its known by different communities under different names like: "Hanna bat Matityahu", "Hanna bat Yochanan", "Yehudit bat Yochanan Cohen Gadol", "Daughter of a Hashmonai", "Yehudit bat Matityahu."
Other: Yael and Sisra and Bat Iftah
The story of Yael and Sisra is told on Hag HaBanot because of its similarities to the story of Yehudit. Here also, Yael seduces Sisra and gets him to sleep with wine and milk, and then kills him.
Dr. Dalia Marcus connects Eid Al-Banat to the story of Bat Iftah as it is another example of a women's day, but was lost to time :(
In Djerba (Tunisia) mostly the single women would celebrate the holiday as a segula to find a successful shidduch
In Tunisia they celebrated on this day a joint bat mitzva for all the girls who turned bat mitzva that year
The Jews of northern Tunisia would give presents to the girls
In the Jewish community of Thessaloniki, Greece, there were traditions similar to those of Yom Kippur. Women would go pray together, recite slihot, and apologize to one another.
In Tunisia there is accounts about one year that the Rabbi gave a sermon on the merit of righteous women and afterwards all the women went in front of the ark and kissed the Torah (wikipedia)