The Shulhan Aruch (Orah Haim 551:2) rules that one may not wed during the Nine Days, however one may get engaged without a festive meal. Although a Seuda is not permitted for an engagement, one may have simple refreshments as are common in what is called a “LeHaim” or “Vort”.
Festive meals are not permitted during the Nine Days in and of themselves, however if they are associated with a Mitzvah, they are permitted. As such, if a boy turns thirteen during the Nine Days, a Seuda may be held in honor of his accepting the yoke of Torah and commandments on his Bar Mitzvah. Conversely, holding a festive meal to dedicate a new home (“Hanukat Bait”) is not permitted during the Nine Days since the dedication itself is not appropriate during this time.
Another common type of festive meal associated with a Mitzvah which is permitted is a Siyum, the completion of a section of the Talmud. One who invested significant time and effort to complete a tractate of the Talmud may hold a festive meal with meat and wine during the Nine Days and the invitees who came to rejoice with him may partake of this meal as well. This leniency has created a situation in which restaurants and the like offer daily or hourlySiyumim so that meat could be served in their establishments. This is clearly not in the spirit of the law nor is it appropriate given the mournful nature of this period. Indeed, Rabbi Ben Zion Abba Shaul (Or LeTzion vol. 3, 26:5) questions the propriety of partaking of such a meal by people who are not connected to the person that is holding the Siyum. Furthermore, he mentions that once one has ten participants for the Siyum, one should not invite more people. Therefore it is questionable if such a meal is considered a Seudat Mitzvah for someone who is not connected to the Siyum (such as family or a study partner) and as such eating meat might not be permitted.
Summary: Festive meals not associated with a Mitzvah are not permitted during the Nine Days.