I guess the best way to answer this is to go around the Jewish calendar. Please keep in mind that this is an Orthodox Jewish response, and not every Jew will follow all of these holidays. Specifically, the Reform and Conservative Jews will not follow some of them.
The Jewish year begins with Rosh Hashana. Literally "Head of the Year" or "Beginning of the Year". This is a two-day holiday of prayers for forgiveness for the sins of the prior year. We spend quite a bit of this Holiday in the Synagogue in prayer. There is also a tradition to eat certain foods including apples dipped in honey (symbolizing the sweetness of the new year), fish heads (symbolizing our desire to be at the head, not at the rear of the new year), pomegranites (the number of seeds of the pomegranite are symbolic of the 613 Commandments of G-d). We also blow the ram's horn on this day, which is a symbolic wakeup call for all Jews to begin their attonement. Generally, it is a holiday of introspection and reflection.
This is followed by Tzom Gedalia (The Fast of Gedalia) which takes place the day immediately after Rosh Hashana. This fast commemorates the assasination of one of our great leaders during the Babylonian exile. It lasts from dawn to dusk.
Thereafter, on the 10th day of the year, we reach Yom Kippur, the "Day of Attonement". This a day of intense prayer and fasting in which we beg G-d for forgiveness for our sins. MOST of the day is spent in prayer, and no food or drink is consumed for approximately 25 hours from sundown the night before until one hour after sundown on the day of the holiday.
Five days later is the beggining of an 8-day holiday called Sukkot, known as the Festival of Tabernacles in English. We build what are essentially huts outside our homes and eat our meals in them for 7 days. The last day of the holiday is called "Simchat Torah" or "Celebration of the Torah" and is the annual anniversary of the completion of the cycle of reading there Torah in the Synagogue on the Sabbath. The entire holiday is one of celebration, spending time with family, and good food. In addition to the huts we live in, we also take a palm branch, a citron, a willow branch and a myrtle branch, tie them together and shake them during certain parts of our prayers.
Next along the year is Chanukkah, known as "The Festival of Lights". During this 8-day holiday we light the menorah, adding another candle every night until completion of the holiday. The children play "draidle" which is essentially a top. Commonly eaten foods are potato pancakes and jelly donughts, both of which are oily and symbolize the miracle of the oil for which the holiday takes place.
Next comes another fast day, the 10th of Tevet, which commemorates the beginning of the seige of Jerusalem by Nebuchanazzer. This is a dawn-to-sundown fast.
Then is Tu B'Shvat or the 9th of Shvat, which is a day to celebrate trees and plants and the like. Think of it as a spring festival or sorts. There is a tradition to eat fruits on this day, especially fruits grown in Israel.
The following month is Purim. This festival commemorates the failure of the plot of Haman to destroy the Jews and the victory of Esther and Mordechai over his plans. We celebrate by dressing in costumes, reading the Book of Esther, and trading care packages of foods. We also eat Hamantash (Haman's Hat) cookies.
One month later is Pesach or Passover. I'm sure you've hear of this one. It is an 8-day Holiday that is probably the most complicated of all the Jewish holidays and requires the most preparation. During this 8-day holiday, we cannot eat any leavened products. No breads, cakes or cookies made with wheat flour. In addition, Ashkenazic Jews also don't eat legumes or corn products either. The cleaning process for this holiday is very difficult, and the rules are very strict. The holiday itseld commemorates our leaving Egypt in victory under the leadership of Moses and the end of our slavery under Pharaoh.
40 days later is Shavuot, the Festival of Weeks. We count the days between Passover and Shavuot (7 weeks) until the holiday begins. It celebrates the receipt of the Torah from G-d at Mt. Sinai, which took place 49 days after we left Egypt. During this 2-day holiday, we spend time to study the Torah, and even forgoe sleep the first night in order to study the Torah all night.
Later in the year is the 17th of Tammuz is a dawn-to-sundown fast that commemorates the breach of the Walls of Jerusalem that led to the Destruction of the Great Temple.
Three weeks later, we have the 9th day of Av. This is another fast day, but it is a 25 hour fast, beginning from sundown the night before till one hour after sundown on the day of the fast. No food or water at all, except in the case of medical need. This commemorates the destruction of the Great Temple (both of them), and is the saddest day of the year for Jews. We read the book of Lamentations and say some special prayers of mourning. We refrain from wearing leather shoes, and we sit on low chairs as a mourner would during a Shivah.
And then, less than two months later, we're right back at the beginning of the Jewish year.
Hope this helps.