The 33rd day of counting the Omer (sheaves of wheat). The 50 days from Passover to Shavuouth are counted using sheaves of wheat. This is a period of contemplation before receiving the Torah at Mt. Sinai and usually a time during which there are no celebrations. It is also associated with a plague that killed 24,000 of Rabbi Akiva's students. However, on the 33rd day, none of Akiva's students died, so it became a day of celebration. Also, it is associated with the anniversary of the death of Rabbi Simon Bar Yochai, the author of the Zohar, the original book on Kabbalism (Jewish mysticism). In Israel, bonfires are set in memory of Bar Yochai.
The fifty days between Passover (the Barley harvest) and Shavuoth (the Feast of Weeks, also the Wheat Harvest) are counted by the "omer" -- a sheaf of wheat brought to the Temple as a grain offering after Passover. This is a period of solemnity because of a plague during the time of Rabbi Akiva. It is one of the three Pilgrimage Holidays (Passover and Sukkoth are the other two). The instructions to celebrate Shavuoth appear in the Torah in:
(1) Exodus 23:16 - And the festival of the reaping of the first fruits of your workings that you seeded in the field...
(2) Exodus 34:22 - And the festival of Weeks [Shavuoth] you will make for yourself with the first fruits of the reaping of wheat..
(3) .Leviticus 23:15 - You will count for yourself from the day after the Sabbath from the day of your bringing the Omer [sheaf of wheat] for a wave-offering for seven Sabbaths, complete they will be.
(4) Deuteronomy 16:10 - You will make a Festival of Shavuoth for YHVH your Elohim, a measure of the free-will of your hand
(5) Deuteronomy 16:16 - Three times in the year all of your males will appear before YHVH your Elohim in the place that He chose: with the Festival of the Unleavened Breads, and with the Festival of the Weeks...
Shavuoth became associated with the receipt of the Torah at Sinai by the rabbis. There's no direct reference to this in the Torah, but it is now part of our tradition, celebrated by studying Torah all night. There are no traditional Shavuoth foods per se, although it is customary to have a dairy meal. This, we are told, was because once we had the laws of Kosher in the Torah, our ancestors could not eat the meat they had because it was non-Kosher, so they must have eaten dairy meals until they could kosher appropriate animals.