From Artscroll's Children's Book of Jewish Holidays:

Beginning with the second night of Pesach we count days from the holiday celebrating leaving slavery in Egypt to the holiday celebrating receiving the Torah. The seven weeks of counting are called Sefirah which is Hebrew for counting.

Since this counting begins on the second day of Pesach, the same day the Omer (barley offering) was brought to the Temple, the counting is called "counting the Omer".
We count, "Today is one day in the Omer."
"Today is two days in the Omer."
We continue counting until we reach the forty-ninth day, the end of seven weeks. The day after the forty-ninth day in the Omer is the holiday of Shavuot.

These seven weeks later became a time of mourning, because of the tragedies which occurred during these days. Twenty-four thousand students of Rabbi Akiva died in a plague during the days between Pesach and Shavuot. They were punished because they did not treat each other with proper respect. This should be an important lesson to us. We must always act kindly and respectfully to others.

A thousand years later, during the Crusades in France and Germany, whole communities of Jews were killed during this Omer period. And in the years 1648 and 1649 Bogdan Chmielnicki led Russian Cossacks in the attack and murder of three hundred thousand Jews.

But on the thirty-third day of the Omer there is no mourning. The plague which was killing so many of Rabbi Akiva's students stopped on that day. The number thirty-three in Hebrew is written L"G, which is pronounced Lag, and so the day is called Lag Ba'Omer.

Another famous event happened on the eighteenth of Iyar, the thirty-third day of the Omer. On that day Rabbi Shimon bar Yochai died. He was a great rabbi and teacher. On the day of Rabbi Shimon Bar Yochai's death, he taught his students many of the Torah's hidden lessons. On that day the sun did not set and the day did not end, until he had taught them all that God allowed him to reveal.

Usually the death of a great man is a day of mourning. But Rabbi Shimon bar Yochai wanted it to be a day of celebration, not mourning. He made this wish because of all the Torah he had been able to teach on that day.

Another Lag Ba'Omer custom is that children play with toy bows and arrows. The bows and arrows remind us of when the Romans ruled over the Land of Israel. The Romans did not allow Torah study. Anyone caught studying the Torah was killed. Rabbi Akiva did not stop teaching Torah. He said, "Jews without Torah are like fish without water! We must continue studying the Torah!" He and his students disguised themselves as hunters. They carried bows and arrows deep into the woods. There they would study Torah, sometimes while hiding in caves.

In many communities, very young boys whose hair has never been cut and who have reached the age of three get their first haircuts on Lag Ba'Omer. In the Land of Israel, many people bring their children to Meron, the place where Rabbi Shimon bar Yochai is buried, and there cut their children's hair for the first time. Many thousands of people come to Meron to celebrate Rabbi Shimon bar Yochai's day.