Let’s begin at the beginning. An omer is a tenth of an ephah. These are dry measures that were used in ancient times. An “omer” may also mean a “sheaf” of grain, in this case barley, and that is what we are theoretically counting during these days.
In Leviticus 23:9-11, we will read (in a couple weeks) that we were forbidden to use any of the newly grown grain until an omer of it was taken to the Temple as an offering. The priest would “elevate” it before God on the day after Shabbat on one’s behalf. Then in Leviticus 23:15-16, we see that we are further commanded to count off seven whole weeks – since they must be complete weeks we make it 50 days, and then we would take an elaborate celebratory offering of new grain to the Temple.
Some have said that during the omer period we are partially in mourning – we don’t have weddings, parties, or dinner dances, don’t get haircuts. On the 33rd day – Lag BaOmer (if you write numbers with Hebrew letters, in which aleph = 1, bet = 2, etc., the number 33 creates an acronym pronounceable as “lag,” lamed gimel לג) – we get a break from our mourning because Rabbi Akiva’s students got a break in a plague after tens of thousands died.
Now about that plague, some scholars feel it was really referring to the students who participated in the revolt led by Bar Kochba against the Romans. So as not to become targets of the Romans, they may have shrouded the story of their dead in a mystery disease rather than reveal that they had waged warfare against Rome. One of Rabbi Akiva’s precious few disciples who survived the revolt (or the plague) was Rabbi Shimon bar Yohai, and it is said that later, after continuing to rail against Rome, he had to flee and spend twelve years in hiding with his son Eleazar, and subsequently he died on Lag BaOmer. This was said to be Divine retribution, as he came out of the cave of hiding after twelve years eschewing everything but Torah study, and the Almighty - who had provided sustenance for them - felt that was inappropriate and made him remain becaved an extra year. Despite the punishment, Rabbi Shimon’s asceticism caught on somewhat, and carried forward through Jewish mysticism. He is widely believed to have been an original architect of the Zohar based on Rabbi Akiva’s teachings.
On Lag BaOmer we can dance, light bonfires, take haircuts, get married and whatnot. (Sephardi custom is to celebrate the day following, Lad BaOmer.) In Israel, they go to Rabbi Shimon’s grave near Safed to light fires and sing kabbalistic music. During the Middle Ages, rabbinic students termed Lag BaOmer “Scholar’s Day,” and they participated in outdoor sports.
This year Lag BaOmer begins the evening of April 29th; some say we start celebrating on the morning of the 30th.