The Three Pilgrimage Festivals, known as the Shalosh Regalim, are three major festivals in Judaism - Pesach (Passover), Shavuot (Weeks), and Sukkot (Tabernacles) — when the Israelites living in ancient Israel and Judea would travel to Jerusalem, as commanded by the Torah. In Jerusalem, they would participate in festivities and ritual worship in conjunction with the services of the kohanim ("priests") at the Temple in Jerusalem.
After the destruction of the Temple, the actual pilgrimage is no longer obligatory upon Jews, and no longer takes place on a national scale. During synagogue services the related passages describing the holiday being observed are read aloud from a Torah scroll on the Bimah (platform) used at the center of the synagogue services. During the Jewish holidays in modern-day Israel, many Jews living in or near Jerusalem make an effort to attend prayer services at the Wailing Wall "emulating" the ancient "pilgrimages" in some small fashion.
Sources in the Hebrew Bible
Book of Exodus: "Offer a sacrifice to Me three times each year. Keep the festival of Matzos (that is Passover)...the reaping festival (Shavuot)...the harvest festival (that is Sukkot)... Keep the Festival of Shavuot through the first fruits of your wheat harvest. Also keep the harvest festival [i.e. Sukkot] soon after the year changes. Three times each year, all your males shall thus present themselves before God the Master, Lord of Israel." (Exodus 34:18-23)
Book of Deuteronomy: "...Then count seven weeks for yourself. From the time that you first put the sickle to the standing grain, you must count seven weeks. You shall then celebrate the festival of Shavuot to God your Lord, presenting a hand-delivered offering according to the extent of the blessing that God your Lord has granted you... Three times each year, all your males shall thus be seen in the presence of God your Lord in the place that He will choose: on the festival of matzahs, on the festival of Shavuot, and on the festival of Sukkot. You shall not appear before God empty-handed." (Deuteronomy 16)
After the destruction of the Temple in 70CE the rabbis connected Shavuot with the Revelation at Mt. Sinai, when God gave the Ten Commandments to the Jewish people. This is why Shavuot celebrates the giving and receiving of the Torah in modern times. The Talmud tells us that God gave the Ten Commandments to the Jews on the sixth night of the Hebrew month of Sivan. Shavuot always falls 50 days after the second night of Passover. The 49 days in between are known as the Omer.
The counting of the days and weeks is done in anticipation of the Giving of the Torah (including the Ten Commandments) to the Jewish People. On that historical day they accepted the Commandments or Guidelines for Living and so became a Nation committed to serving God.
In the Torah Shavuot is called Feast of Weeks, Festival of Reaping and Day of the First Fruits.
Three times a year during the three "foot festivals" (shloshet haregalim), Tabernacles (Succoth), Passover (Pesah), and at Pentecost (Shavuot), a prayer used to take place at the Temple. In ancient times, Jewish pilgrims used to travel to Jerusalem from all over the country or abroad on foot. Nowadays, a formal "priestly blessing" (birkat kohanim) is said at the Western Wall in memory of the former pilgrimage. A multitude of people assembles and prays from the early hours of the morning in front of the Wall.