Firstly, Mazal Tov! May Hashem bless your daughter and her chatan that they build a Bayit Neeman b'Yisrael!
The custom of drinking wine under the chupah traces its roots back to very early times - it is first mentioned in the Responsa of the Gaonim over one thousand years ago. The Midrash relates, Bereshet Rabbah 8:13, that when God married Adam and Eve, He took a cup of wine and sanctified the union. The custom is also mentioned in the Zohar, see 2:169, as the sanctification of the chupah is similar to that of Shabbat. It is also based on a Talmudic dictate that wine is a symbol of joy for both humans and God, see Tractate Brachot 35a, and it is codified in the Code of Jewish Law.
Because the blessing is recited over the cup of wine some of it must be drunk so as not to have caused a blessing to have been said in vain. The custom is for the chatan and kalah to be the ones to drink the wine as they are the ones that the blessing was recited for. It is very symbolic that, as they start off their married life together, they drink from the same cup.
For more information about this and the wedding ceremony in general I recommend a book called Made in Heaven by Rabbi Aryeh Kaplan published by Moznaim Publishing Corporation.
