What is Sefirat HaOmer?
(15) And from the day on which you bring the sheaf of elevation offering—the day after the sabbath—you shall count off seven weeks. They must be complete: (16) you must count until the day after the seventh week—fifty days; then you shall bring an offering of new grain to the Lord.
Who was Rabbi Akiva?
"ואהבת לרעך כמוך" – רבי עקיבא אומר זה כלל גדול בתורה. בן עזאי אומר "זה ספר תולדות אדם" – זה כלל גדול מזה.
"And you shall love your neighbor as yourself" Rabbi Akiva says: This is an all-embracing principle in the Torah. Ben Azzai says: (Genesis 5:1) "This is the numeration of the generations of Adam" — This is an even greater principle.
Customs of Sefirat HaOmer
(א) נוֹהֲגִים שֶׁלֹּא לִשָּׂא אִשָּׁה בֵּין פֶּסַח לַעֲצֶרֶת עַד ל''ג לָעֹמֶר, מִפְּנֵי שֶׁבְּאוֹתוֹ זְמַן מֵתוּ תַּלְמִידֵי רַבִּי עֲקִיבָא; אֲבָל לְאָרֵס וּלְקַדֵּשׁ, שַׁפִּיר דָּמִי, וְנִשּׂוּאִין נָמֵי, מִי שֶׁקָּפַץ וְכָנַס אֵין עוֹנְשִׁין אוֹתוֹ. הַגָּה: מִיהוּ מִלַּ''ג בָּעֹמֶר וָאֵילָךְ הַכֹּל שָׁרֵי (אַבּוּדַרְהַם וּבֵית יוֹסֵף וּמִנְהָגִים).
(ב) נוֹהֲגִים שֶׁלֹּא לְהִסְתַּפֵּר עַד ל''ג לָעֹמֶר, שֶׁאוֹמְרִים שֶׁאָז פָּסְקוּ מִלָּמוּת.
(1) 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 OK. 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).
(2) It is customary not to cut one's hair until Lag BaOmer, since it is said that that is when they stopped dying.
What happened to his 24,000 students?
אמרו שנים עשר אלף זוגים תלמידים היו לו לרבי עקיבא מגבת עד אנטיפרס וכולן מתו בפרק אחד מפני שלא נהגו כבוד זה לזה והיה העולם שמם עד שבא ר"ע אצל רבותינו שבדרום ושנאה להם ר"מ ור' יהודה ור' יוסי ורבי שמעון ורבי אלעזר בן שמוע והם הם העמידו תורה אותה שעה תנא כולם מתו מפסח ועד עצרת אמר רב חמא בר אבא ואיתימא ר' חייא בר אבין כולם מתו מיתה רעה מאי היא א"ר נחמן אסכרה.
It was said, That Rabbi Akiva had twelve thousand pairs of disciples, from Gabbat to Antifaras; and they all died at the same time because they did not treat each other with respect. The world remained desolate until Rabbi Akiva came to our Masters in the South and taught the Torah to them. and it was they who revived the Torah at that time. These were Rabbi Meir, Rabbi Judah, Rabbi Yossi, Rabbi Shimon and Rabbi Eleazar ben Shammua; And it was they who revived the Torah at that time. A Tanna taught: All of them died between Passover and Shavuot. Rabbi Hama bar Abba or, it might be said, Rabbi Chiyya bar Abin said: All of them died a cruel death. What was it? — Rabbi Nahman replied: (diptheria or croup.)
אמר להם הראשונים לא מתו אלא מפני שהיתה עיניהם צרה בתורה זה לזה אתם לא תהיו כן מיד עמדו ומלאו כל ארץ ישראל תורה
[Rabbi Akiva said to his new students:] the first students died because their eyes were narrow in the Torah they shared with each other. Do not be like this. They immediately stood up and filled the land of Israel with Torah.
עץ יוסף, עין יעקב, יבמות סב ב מפני ושם במדרש איתא מפני שהיתה עיניהם צרה אלו לאלו כלומר שהיו שונאים זה לזה ולא היו רוצים לההנות זה לזה בלמוד וכאן פירש שלא נהגו כבוד כו' שלא חש כל אחד מהם על כבוד התורה של חברו דאין כבוד אלא תורה
Etz Yosef on Ein Yaakov Yevamot 62b
And as it is brought in the Midrash (Kohelet Rabbah), "because their eyes were narrow with the Torah they shared with each other." This it to say, that they hated one another, and didn't want to enjoy each other's company in Torah learning, and they weren't respectful with one another, etc., as they were not careful to be respectful of the Torah of their fellow, as there is no such thing as honor except in regards to Torah.