Rabbi Yochanan Eskenazi
During the Nine Days, it is prohibited to eat meat or drink wine (Shulchan Aruch 551:9). One reason is because it is inappropriate to indulge during this intense national mourning period (Aruch Hashulchan 551:23). A second reason is that since the time of the destruction of the Second BaisHamikdash, ideally one should refrain from eating meat and drinking wine completely, since we can no longer offer karbanos [meat] and bring wine libations. However, since it would be too hard to always refrain from these foods, Chazal only decreed not to for a short period of time [the Nine Days] as a reminder of what we are missing (Gemara Bava Basra 60b) (Gra 551:9). Included in this prohibition is one should refrain from any meat [including fowl] and food that was cooked with any of the above mentioned items. If one mistakenly recited a brachah on meat or wine, etc. he should partake a tiny amount [to avoid saying a brachahl'vatalah] (Laws of Daily Living pg. 54). Grape juice is considered wine, and may not be consumed (Laws of Daily Living pg. 53 ftnt. 15 quoting Harav Moshe Feinstein, zt"l). One may drink beer (Rama 551:11).
Certain people are not included in this prohibition. One who is sick [even if his illness is not life threatening] (Mishneh Berurah 551:61), or someone who is unable to eat dairy foods may eat chicken [but not beef] until Erev Tisha B'Av (Mishneh Berurah 551:64). Pregnant women may eat chicken. If no chicken is available, it is permitted to eat beef (Aruch Hashulchan 551:61). Additionally, a nursing woman [if by refraining from meat will be detrimental to her milk] may eat beef (Mishneh Berurah 551:64). One may eat meat or drink wine at a seudas mitzvah [including a bris, pidyon haben, siyum, & bar mitzvah] (Rama 551:10).
It is well known, if one partakes in a siyum, one is permitted to eat meat. The Maharshal [Yam Shel Shlomo, Bava Kamma, end of seventh perek] writes, "There is no greater simcha or mitzvah that is done before Hashem than the simcha and mitzvah of finishing a portion of Torah." Even a child who completed and understood what he learned may make a siyum which adults may partake in (Laws of Daily Living pg. 59 quoting Shu"T B'tzail Hachochmeh 4:100). However, one should not leave over part of a mesechta in order that he should be able to make a siyum during the Nine Days (Mishneh Berurah 551:73). If this did happen, it is permissible to partake in the siyum (Koveitz Halachos 9:ftnt. 33). As an aside, many ChassidishRebbes would encourage their Chassidim to make a siyum specifically during the Nine Days, for it was hoped that study of the Torah and joyous celebration over the learning of Torah, will help bring the geulah shelaima (Laws of Daily Living pg. 59).
One may eat meat and drink wine on Shabbos. This includes even if he accepts upon himself early Shabbos, and does not end Shabbos until after the zman on MotzaiShabbos (Mishneh Berurah 551:56).
There is a machlokes whether the one reciting havdala may drink the wine. Some opinions hold that the adult reciting the havdala should not drink the wine but rather give it to a minor (Rama 551:10) who has reached the age of chinuch, but is not old enough to understand mourning (Mishneh Berurah 551:70). Other opinions hold that one should recite havdala on chamar medina [e.g. beer] (Aruch Hashulchan 551:26). Harav Moshe Feinstein, zt"l (Laws of Daily Living pg. 65 ftnt. 63) and ybl"c Harav Shmuel Kamenetsky, shlit"a (Koveitz Halachos 9:19) hold that one may l'chatchila recite havdala on wine or grape juice and drink it himself, even when a kattan is present.