When: Rosh Hashanah is observed on the first and second of Tishrei. The name means "Head of the Year." Its Biblical names are Yom HaZikkaron (the day of remembrance) and Yom Teruah (the day of the sounding of the Shofar, which is usually a ram's horn but can come from any kosher animal except the cow).
The observance is instituted in Leviticus 23:23-25
Various Studies:

Tzom Gedaliah (the Fast of Gedaliah) is an annual fast day instituted by the Jewish Sages to commemorate the assassination of Gedaliah Ben Achikam, the Governor of Israel during the days of Nebuchadnezzar King of Babylonia. As a result of Gedaliah’s death the final vestiges of Judean autonomy after the Babylonian conquest were destroyed, many thousands of Jews were slain, and the remaining Jews were driven into final exile. The fast is observed on the day immediately following Rosh Hashanah, the third of Tishrei. In the Prophetic Writings this fast is called 'The Fast of the Seventh' in allusion to Tishrei, the seventh month. Various Studies:

When: Sukkot begins on Tishrei 15

Sukkot was one of the three annual pilgrimages to the Jerusalem Temple (Exodus 34:22). It marks the end of harvest time and thus of the agricultural year in the Land of Israel. See Leviticus 23:33.

When: Observed on the 22nd and 23rd day of Tishrei 22 (which is the eighth day after entering the sukkah).

This is a Rabbinic Tradition derived from Torah. The festivals mark the completion of the annual cycle of Torah readings (or parsha) and the beginning of the annual reading cycle (with Bereshit/Genesis 1:1). Two days of celebration with dancing and joy. Often viewed as part of Sukkot, but technically occuring on the days following it.
We now return to normal yearly cycle of our observances.

This Shabbat is known as Yud-Tes Kislev - "Rosh Hashanah of Chassidism" - because of the yartzeit of Rabbi DovBer, known as "The Maggid of Mezeritch." He was the disciple of, and successor to, the founder of Chassidut, Rabbi Israel Baal Shem Tov. Rabbi DovBer led the Chassidic movement from 1761 until his passing on Kislev 19, 1772.

Mordecai the Jew recorded the events found in the Megillah of Esther. There we learn that he sent dispatches to all the Jews throughout the provinces of King Ahasuerus, near and far, charging them to observe the fourteenth and fifteenth days of Adar, every year — the same days on which the Jews enjoyed relief from their foes and the same month which had been transformed for them from one of grief and mourning to one of festive joy. Jews observe the day as a time of feasting and merrymaking, and as an occasion for sending gifts to one another and presents to the poor.

Purim remembers the defeat of a plot to exterminate the Jews of Persia (modern Iran). In a greater sense it reminds us of all the innumerable attempts to destroy us throughout our generations. It also reminds us of HaShem's constant protection. Purim is a day of great joy and hence has been called the Jewish Halloween due to the costumes, public celebrations, and considerable alcohol consumption. As we recite each year in the Pesach Haggadah, "In each and every generation the Gentiles rise up against us to destroy us. But the Holy One, blessed be He, rescues us from their hands." On Purim we celebrate His continuing protection.

Purim Katan
Ta'anit
Fast of Esther
Purim and Shushan
Purim Spiels
And more.

The Passover history is recorded in Exodus Chapters 1-15. Many of the specific Pesach observances are instituted in Chapters 12-15, other aspects come from diverse Jewish tradition.
During Pesach all chametz (leaven: anything made from the five major grains -- wheat, rye, barley, oats and spelt) -- are removed from the home and strictly avoided. Ashkenazi Orthodox Jews also consider rice, corn, peanuts, and legumes (beans) to be of this category by their tradition. Everything regarded as chametz is removed from the home in memory of the fact that the Israelites fled Egypt in a hurry -- not leaving time to even let their bread rise. This tradition is also symbolic of our desire to remove the puffiness of arrogance and pride from our souls. This is a time to take stock of what really matters in life realizing that in a moment our lives could be drastically overturned by circumstances beyond our control. Our only hope is in HaShem alone. In His alone we place our emunah (motivating faith).

Chassidut:
Lag B'Omer is the heart of the Omer Count connecting Pesach (Exodus from slavery) to Shavu'ot (freedom from slavery through the revelation of HaTorah). The Torah commands counting the days from Pesach to Shavuot. This period is known as Sefirat HaOmer (the Counting of the Omer, see above entry). On Lag B'Omer the various period rules are suspended for a single day of joy and the rememberance of the great Rabbi Shimon bar Yochai (aka "Rashbi").

It is an ancient tradition to stay up all night on the first night of Shavu'ot and give oneself to Torah and Talmud study. As the morning dawns one should be engaged in Torah and prayer. It is also a tradition to read the Book of Ruth during the night. Dairy meals are also part of the traditional observance. Work is not permitted on Shavu'ot.

A day for remembering diverse major Israelite tragedies, although Tisha B'Av primarily (especially) commemorates the destruction of the first and second Temples (both were destroyed on the ninth of Av: the first by the Babylonians in 586 B.C.E.; the second by the Romans in 70 C.E.). On this day the Torah cabinets are draped in black cloth.
Begins with the fast of the 17th of Tammuz (commemorating the first breach in the walls of Jerusalem, before the First Temple was destroyed. During this period worldly pleasures are increasingly avoided, parties, cutting of the hair etc are not permitted. From the first to the ninth of Av tradition says to refrain from eating meat or drinking wine (except on Shabbat) and from wearing new clothing.

When: The entire month

Jewish Tradition reserves the month of Elul for preparation for the High Holy Days:

The 1st of Elul is the New Year's Day for the tithing of cattle. When the Temple is standing the tithe for cattle is made from cattle born in the same fiscal year, i.e. between the 1st of Elul one year to the next.
Elul is, more importantly, the month of preparation for the High Holy Days. It is said that during this month HaShem leaves His Holy Place to walk among us looking for souls to forgive and restore. Elul is therefore the month of teshuvah or repentance. It is a time for intensive introspection and soul correction. During Elul we are encouraged to clarify our goals and plans for the upcoming year and beyond. More importantly however, this is the time for renewed spiritual communion and for coming closer to G-d with devekut (attachment) through increasing our emunah (active faith). Elul is a time for realizing ones purpose in life. During Elul ones attention forsakes sense gratification and turns to spiritual pursuits. We are more aware that we will soon stand before the Judge of all judges and we prepare our cases by seeking to correct all uncorrected wrongs.
The shofar is sounded each day of Elul. Some Poskim rule one is to begin to blow the Shofar from the first day of the Elul Rosh Chodesh (being the 30th of Av) while other Poskim rule the blowing of the Shofar begins the second day of the Rosh Chodesh (being the first of Elul).
Our sages explain that the four Hebrew letters of the word Elul (aleph-lamed-vav-lamed) are the first letters of the four words אֲנִ֤י לְדוֹדִי֙ וְדוֹדִ֣י לִ֔י הָרֹעֶ֖ה בַּשׁוֹשַׁנִּֽים׃ (ס) — "I am my beloveds" and my beloved is mine (Song of Songs 6:3). During the month of Elul HaShem abides with His people in special intimacy.

Rosh Chodesh marks the first day of any new month.
In biblical times this was a significant holiday that included the sounding of the shofar (ram's horn). Today observance mainly consists of slight changes in the daily prayer recitations such as the recitation of the birkat ha-hodesh (a special prayer for the month to come, from peace and prosperity to success in business, good health, and piety). One should not fast on Rosh Chodesh
It is good to eat well (with bread) on this day.
No cutting of hair, nails etc.
Some Jewish women still take the day off in memory of their refusal to participate in the incident of the Golden Calf:

"And Aaron said: Take the earrings from your wives, sons and daughters, and bring them to me" (Exodus 32:2). The women heard and refused to give their jewelry to their husbands, but said: "You want to make a calf with no power to save? We will not listen to you." God gave them reward in this world that they keep Rosh Chodesh more than men, and in the next world they merit to renew themselves like Rosh 1Chodesh. (Pirkei d'Rabbi Eliezer, 45)