Remembering the Destruction of Jerusalem
3. [As a reminder of the destruction of the Beit Hamikdash (Temple), the Sages ] also ordained that one should not listen to any live musical instruments, and even vocals (1). At festive meals, all that should be sung are the traditional "zemiros" ("songs of praise [for G-d]") that were specifically instituted for the occassion, for example, those sung at the Shabbos table (2). Other songs may not be sung (3).
(1) According to the Ramah (560:3), this prohibition applies only to people like past kings and princes, who had live music played constantly for them in the palace. According to this opinion, the prohibition also applies to any situation in which people are celebrating with wine ("Beit Ha'Mishteh") while listening to the music or singing, even people who do not regularly have live music played for them; in other words, regular people (as opposed to kings, as above) may play and listen to live music, as long as wine is not being drunk at the time (See Mishna Berura 560:13 for an alternative opinion).
(2) In other words, only songs of praise for G-d may be sung or listened to, while drinking wine. These songs are permissible even at meals that are not associated with a mitzvah (Shulchan Aruch 560:3). As we shall see in HY 126:4, all forms of live music and song are permissible when a mitzvah is involved, such as a wedding.
(3) That is, while people are drinking wine (See Mishna Berura 560:13 for an alternative opinion).