The Torah tells us that after Rivkah agreed to return with Avraham's servant to marry Yitzchak, her family gave her a Beracha as they sent her away (בראשית כ"ש:ס). The Beraisa at the beginning of the first Perek of Maseches Kallah states that this Posuk is the source of our practice to recite the Birchas Chassanim, better known as the Sheva Berachos, at a wedding. We likewise read in Pirkei D'Rabbi Eliezer (פרק ט"ז) that they stood and gave Rivkah a Beracha just as a Chazan stands and recites Berachos in the presence of the Kallah under the Chupah. The Gemara there in Maseches Kallah as well as in Kesubos (שף ז:) cites other Pesukim which similarly hint at the practice to recite these special Berachos.
The Mishnah in Megillah (דף כ"ג:) indicates that a Minyan of ten men is required in order to say these Berachos, the aforementioned Gemara in Kesubos derives these points from the Pesukim as well. On the next page in Kesubos (דף ח.), the Gemara states that the Chassan himself may be counted toward the Minyan. Rabbi Pinchas HaLevi Horowitz, in his Sefer Hafla'ah on Kesubos (שם בד"ה כל זה) suggests that this may depend upon which Posuk is the actual source for requiring a Minyan altogether; the Shulchan Aruch, however, (אבן האזר סימן ע"ב סעיף ד`) rules clearly that the Chassan does count. The Netziv, in his commentary to the Sheiltos (העמה שאלה לשאילתא ט"ז אות י"א), explains that the Chassan can be part of the Minyan because he too has an obligation to be "משמח" the Kallah, which apparently is accomplished by saying these Berachos. However, the Mordechai in Kesubos (סימן קל"א) quotes from one of the Geonim that it is preferable that the Chassan himself not be the one to actually recite the Berachos.
The Gemara earlier in Kesubos (דף ז:) states that these Berachos are recited not only at the wedding, but at every meal throughout the Shivas Yemei HaMishteh, that is, the full week of celebration which follows the actual wedding. This too, though, is only if a Minyan is present. The Pischei Teshuvah (אבן האזר ס"ק ח`) discusses whether the ten men must all eat at the meal or whether it's enough that the ten men are present; he concludes that a majority of them must have eaten at the meal. There are, however, other opinions on this matter; the more prevalent custom is that all ten should eat, at least seven should have bread, while the other three should have a bite of something (עיין ספר הנשואין כהלכתם, חלק ב` פרק י"ד סעיף מ"א).
The Gemara (דם) quickly adds, though, that even with a Minyan, the Sheva Berachos are recited for a full week only if Panim Chadashos, new faces, are present at each meal throughout the week. The Rambam (פרק ב` מבל` ברכות בלכ` ט,י) and the Shulchan Aruch (שם סעיפום ה`-ח`) rule accordingly. It is worth noting that there is a dispute as to who qualifies as one of the required Panim Chadashos. The Rambam (שם הל` י) understands that it must be someone who did not hear the Berachos recited at the wedding, while the Rosh in Kesubos (פרק א` סימן י"ג) writes that it may be someone who did not partake of any of the festive meals yet, even though he may have heard the Berachos recited at the wedding. The Shulchan Aruch (שם סעיף ז`) brings both views expressing preference for the latter. The Beis Shmuel (שם ס"ק ח`), however, that the Maharashal accepted the first view, that of the Rambam.
The Rosh (שם) adds that the new person coming as the Panim Chadashos must be someone whose presence really enhances the joy, that is, someone on whose account there will be even more "שמחה" at the meal. This point is also quoted in Tosafos in Kesubos (שם בד"ה והוא שבאו), and is brought down in the Shulchan Aruch (שם סעיף ח`) as well. However, the Shittah Mekubetzes in Kesubos (שם בד"ה והוא שבאו) quotes from Rashi that it is not necessary to have any special person, and that anybody who attends can serve as the Panim Chadashos. This seems to be the prevalent practice today, although it may be preferable to follow the first view and invite a more distinguished guest, if possible (עיין ספר הנשאין כהלכתם שם סעיף ס"ט). The Ramo (דו"ת שם סעיף ז`) notes that the person serving as the Panim Chadashos need not eat anything at the meal; the Ritva in Kesubos (שם בד"ה ת"ר) writes that he may come even after the meal is over, as long as he is there for the Sheva Berachos.
The above cited Tosafos in Kesubos and many other Rishonim write that on Shabbos, the Sheva Berachos may be recited even without any Panim Chadashos present; the Shulchan Aruch (שם סעיף ח`) accepts this view and extends it to Yom Tov as well. The reason suggested in Tosafos (שם), and elsewhere is that based on a Midrash, the Shabbos itself is considered to be Panim Chadashos, and there is certainly additional joy at the meal on Shabbos (and Yom Tov). The Shittah Mekubetzes, (שם בסוף הדיבור), however, quotes the Ramban that it is because there are always additional people around on Shabbos. The Chasam Sofer, in his Chidushim on Kesubos (שם בד"ה במקהלות), writes that a woman can serve as a Panim Chadashos if her presence enhances the gladness. The aforementioned Ritva (שם), however, clearly says she cannot, because the Panim Chadashos must be someone who can count toward the Minyan required for these Berachos. This question may relate to the function of the Panim Chadashos as well as to why Shabbos counts as Panim Chadashos, but the latter view appears to be the accepted practice, as noted in the Pischei Teshuvah (שם ע"ק י"ד).