Starting on the second day of Passover, the Torah (Leviticus 23:15) says it is a mitzvah every day to "count the Omer" -- the 50 days leading up to Shavuot. This is an important period of growth and introspection, in preparation for the holiday of Shavuot which arrives 50 days later.The first 33 days:
Shavuot is the day that the Jewish people stood at Mount Sinai to receive the Torah, and as such required a seven-week preparation period. The commentators say that we were freed from Egypt only in order to receive the Torah and to fulfill it. Thus we were commanded to count from the second day of Pesach until the day that the Torah was given -- to show how greatly we desire the Torah.
The Talmud tells us that Rabbi Akiva had 24,000 students who tragically died during the Omer period, because they did not treat each other with sufficient respect. Therefore, for the 33 days from Passover until Lag B'Omer, we observe these signs of mourning:
1) no weddings
2) not listening to instrumental music, either live or recorded (vocal music is permitted)
3) no haircuts or shaving, unless for business purposes