Lag B'Omer is the 33rd day of the Omer count, marking a break in the mourning period observed between Passover and Shavuos. It commemorates the passing of Rabbi Shimon bar Yochai, the revered sage and mystic, and the cessation of the plague that afflicted Rabbi Akiva’s talmidim.

Chazal say that Reb Akiva's students were punished שלא נהגו כבוד זה לזה, "because they didn't act with honor towards one another."
There are several customs pertaining the mourning practices of sefirah. The general custom is not to take haircuts, not to hold weddings, and not to buy new clothes in order not to recite Shechiyanu, for 33 days. Some have the custom of starting from the beginning of the sefirah count for the first 33 days. Others refrain starting Rosh Chodesh Iyar until 3 days before Shevuos. Others refrain the entire period between Pesach and Shevuos.

The most common custom is to refrain from celebratory practices for the first 33 days, since the plaque stopped on the 33rd day.
Why specifically the 33rd day?
Rav Meilech Biederman offers the following explanation: The forty-eight kinyanim (acquisitions) of Torah (mentioned in Pirkei Avos ch.6) allude to the first 48 days of the sefirah (and the 49th day is a combination of them all). The 32nd acquisition is אוהב את הבריות, to love people. When one attains that level, he will honor his fellow man. This can explain why the students stopped dying on the 33rd day (ל''ג בעומר). After they reached the level of loving their fellow man, they honored them properly.