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 [Note: According to some Ashkenaz customs, the 33-day mourning period begins a few weeks later on the first of Iyar, and ends on the third of Sivan.]:

No weddings
Not listening to instrumental music, either live or recorded (vocal music is permitted)
No haircuts or shaving, unless for business purposes

According to the Ashkenazi custom, these restrictions apply until the 33rd day of the Omer and according to the Sephardic custom, until the morning of the 34th day of the Omer. Some Sephardic individuals act leniently with regards to haircuts beginning from the 33rd day of the Omer. Those who act leniently in this regard have on whom to rely.

If possible a man should abstain from shaving during the Omer period as well. However, there are those who rule leniently for individuals who are truly distressed as a result of not shaving their beard, for the Radbaz writes regarding such matters which are not actual obligations as a result of an edict of our Sages and is merely a custom, in a case of such distress, there is room for leniency. (It is especially worthy to be stringent until Rosh Chodesh Iyar).

For Sephardim, women are not included in the prohibition of taking haircuts during the days of the Omer, for even with regards to actual mourning for a relative who has passed away (for which a male mourner must abstain from taking a haircut for the entire thirty-day mourning period), Maran Ha’Shulchan Aruch rules that women are not included in this prohibition and are permitted to take haircuts during the thirty-day mourning period.