AND HE SHALL WAVE [THE OMER] — Every form of the root נוף used in connection with sacrifices signifies that one moves the latter about both horizontally (מוליך ומביא) and upwards and downwards (מעלה ומוריד); he moves it about horizontally to prevent (i. e. the act was symbolical of preventing) destructive winds, he moves it about upwards and downwards to keep away injurious dews (cf. Rashi on Exodus 29:24) (Menachot 62a).
Anonymous author, Spain, 13th century
The commandment of counting the omer: To count forty-nine days from the bringing of the omer which is on the sixteenth day of Nissan, as it is stated (Leviticus 23:15), "And you shall count for yourselves from the morrow of the Shabbat from the day of your bringing the omer of waving." And this tallying is an obligation, and it is upon us to count the days on each day and, likewise, the weeks. As Scripture said to count fifty days and also said (Deuteronomy 16:9), "Seven weeks shall you count for yourself." And in explanation, Abbaye said in the Talmud in Menachot 66a, "It is a commandment to tally the days and it is a commandment to tally the weeks".
The basis of the commandment -- from the angle of the simple understanding [that it is] since the entire essence of Israel is only the Torah, and for the Torah the heavens and earth were created, and as it is stated (Jeremiah 33:25): "Were it not for my covenant day and night, etc." And it is the essence and the reason that they were redeemed and left Egypt - in order that they receive the Torah at Sinai, and fulfill it. And as God said to Moses (Exodus 3:12): "And this will be the sign for you that I have sent you; when you take out the people from Egypt, you shall worship God on this mountain." And the understanding of the verse is [that] your taking them out from Egypt is a sign for you that you shall worship God on this mountain - meaning that you shall receive the Torah, which is the great reason for which they were redeemed and it is their ultimate good. And it is a great matter for them, more than freedom from slavery. And therefore God made a sign for Moses when they left bondage to receive the Torah, as what is secondary will forever be a sign for the essential. And because of this - for it [the Torah] is the essence of Israel, and for its sake they were redeemed and received all the greatness they would receive - we were commanded to count from the morrow of the first day of Passover until the day of the giving of the Torah, to display our great desire for the day we honor and yearn for, like 'a slave seeking shade' (Job 7:2) and constantly counting the time until he reaches freedom, for all he yearns for is that time.
If a man married a woman in his youth, and she passed away, he should marry another woman in his old age. If he had children in his youth, he should have more children in his old age, as it is stated: “In the morning sow your seed, and in the evening do not withhold your hand; for you do not know which shall prosper, whether this or that, or whether they both alike shall be good” (Ecclesiastes 11:6). This verse indicates that a man should continue having children even after he has fulfilled the mitzva to be fruitful and multiply. Rabbi Akiva says that the verse should be understood as follows: If one studied Torah in his youth he should study more Torah in his old age; if he had students in his youth he should have additional students in his old age, as it is stated: “In the morning sow your seed, etc.” They said by way of example that Rabbi Akiva had twelve thousand pairs of students in an area of land that stretched from Gevat to Antipatris in Judea, and they all died in one period of time, because they did not treat each other with respect. And the world was desolate of Torah until Rabbi Akiva came to our Rabbis in the South and taught his Torah to them. This second group of disciples consisted of Rabbi Meir, Rabbi Yehuda, Rabbi Yosei, Rabbi Shimon, and Rabbi Elazar ben Shamua. And these are the very ones who upheld the study of Torah at that time. Although Rabbi Akiva’s earlier students did not survive, his later disciples were able to transmit the Torah to future generations. With regard to the twelve thousand pairs of Rabbi Akiva’s students, the Gemara adds: It is taught that all of them died in the period from Passover until Shavuot. Rav Ḥama bar Abba said, and some say it was Rabbi Ḥiyya bar Avin: They all died a bad death. The Gemara inquires: What is it that is called a bad death? Rav Naḥman said: Diphtheria.
It is customary not to get married between Pesach and Shavuot, until Lag BaOmer (the 33rd day), because during that time, the students of Rabbi Akiva died. However, to do "erusin" and "kiddushin" (engagement and betrothal) is fine. And even for "nisuin" (marriage), if someone did so, we do not punish him. Rema: however, from Lag Ba'Omer onwards, all this is permitted (Abudraham, Beit Yosef & Minhagim).
It is customary not to cut one's hair until Lag BaOmer, since it is said that that is when they stopped dying. One should not cut one's hair until the 34th day, in the morning, unless the 33rd day falls on Friday, in which case one may cut one's hair then, in honor of the Sabbath ("kavod Shabbat"). Rema: But in these countries, we do not follow the custom he advocates; rather, we cut our hair on the 33rd day, and we rejoice a bit, and we do not say Tachanun (Maharil and Minhagim). And one should only cut one's hair on the 33rd day itself, but not on its evening. However, if it falls on Sunday, our custom is to cut our hair on Friday in honor of the Sabbath (Maharil). Someone who is performing a circumcision (i.e., the sandak, mohel and the infant's father), or circumcizing his son, is allowed to cut his hair during the counting of the Omer in honor of the circumcision (Hagahot Minhagim).