It is customary among the entire Jewish nation not to hold weddings during the days of the counting of the Omer, from Pesach until the 34th day of the Omer.
There is a source quoted for the custom of abstaining from getting married during this period in the Responsa of the Geonim (who were the Jewish Sages who lived before the period of the Rishonim), for this is a custom of mourning, as our Sages tell us (Yevamot 62b) that Rabbi Akiva had twelve-thousands pairs of students who all died between the holidays of Pesach and Shavuot because they did not treat each other respectfully. They all died from an illness called “Askara” (an agonizing illness which causes acute respiratory distress and stoppage of breathing).
The reason why we customarily abstain from these things until the 34th of the Omer is based on the words of the Sefer HaManhig (page 72b, authored by Rabbeinu Avraham bar Natan HaYarchi, the Ra’avan, who lived in Lonil and passed away in the year 4975 [1215]) in the name of Rabbeinu Zerachya HaLevi (the Razah, Ba’al HaMaor) who found in an early manuscript unearthed in Spain, that all of Rabbi Akiva’s students passed away from Pesach until “Half of Shavuot,” and this refers to half of the period prior to Shavuot, which is thirty days, as the Mishnah tells us, “One must begin to inquire about the laws of Pesach thirty days before the holiday”; thus, half of thirty is fifteen days before the holiday of Shavuot. Several other Rishonim write likewise. Thus, if we subtract fifteen from the forty-nine days between Pesach and Shavuot, the difference is thirty-four. Nevertheless, beginning from the morning of the 34th day of the Omer, it is already permissible to hold weddings, for regarding mourning, the rule is, “A portion of the day is considered like the entire day,” and since some of the 34th day has already passed, one need not continue to observe the mourning customs.
The Ashkenazi custom is to begin permitting weddings from the 33rd day of the Omer, for they are of the opinion that the students of Rabbi Akiva ceased dying on the 33rd day of the Omer, as several Rishonim (among them the Sefer HaManhig) write that they have a tradition that they ceased dying on the 33rd day. The Rama (Chapter 493) indeed rules this way. Some Ashkenazim are even more lenient and allow weddings to be held on the night of the 33rd of the Omer.
An engagement party marking a match being made, a.k.a. “Tena’im,” may be held during the period of the counting of the Omer, even with musical accompaniment. However, regarding an engagement party held after the match was already set and the couple has decided to marry, it is permissible to hold such a party during these days, albeit without musical accompaniment.