Name of the Month
"Sivan" is the Babylonian name of this month, as are all of the "official" names of the months in the Hebrew Calendar. In the Bible, the month is referred to as "the Third Month," with reference to Nisan, the First Month.
Zodiac Sign of the Month
The "sign" of the month is "Twins," a hint of the roles in this month of Moshe and Aharon who were considered equal, and it was through them and by their merit that the Torah was given to the People of Israel.
Chodesh Sivan - In Relation to Other Months of the Year
2 Iyar (29 days)
3 Sivan (30 days)
4 Tammuz (29 days)
"Rabbi Yossi HaGelili taught in the presence of Rav Chisda (Shabbat 88a),
'Blessed is our G-d Who gave us a Torah which is divided into three parts:
The Five Books of Moses
The Books of the Prophets
The Sacred Writings
To a People that is divided into three parts:
"Kohanim" (the Priestly Component)
"Leviim" (the Levite Component)
The significance of being a member of the "Kohen" or "Levi" Classes of the People of Israel was mainly evident in the time of the Temple, and will hopefully be evident soon, in our time. But significant honors are accorded members of these classes even nowadays, such as in the order of being "called up" to the Torah, and who leads the Grace After Meals, etc.
Yisraelim (the Israelite Component ("plain old" Jews, members of the People which is considered a "Kingdom of Priests and a Holy Nation," by virtue of its observance of the Laws of the Torah))
Through a person who was himself a "Third" - Moshe Rabbeinu, Moses our Teacher, who was the third child born to his mother, Yocheved, in Mitzrayim, after his elder siblings, Aharon and Miriam.
After another "three days of Preparation," they were to be ready to receive the Torah, as it is written, when Moshe commanded the Jewish People to "Make yourselves ready by the third day."