http://revach.net/article.php?id=3803

Section: Hanhagos   Category: Minhagim
Shavuos: The Name Shavuos - Tna'im, Kesuba, And Wedding Vows
Why don't we call the Yom Tov Shavuos, Chag Matan Torah?  What is the significance of the name Shavuos?  The Chida says that Shavuos means to swear.  It was the day that Hashem swore we would be His nation and he would never trade us for another.  We also swore to Hashem that w would never trade Him for another.

Just like when a Choson and Kallah get married they read the Tna'im the promises that each side makes to the other, so too on the first day of Shavuos some have a Minhag (Minhag Yisroel Torah) to read the Tna'im between Hashem and Am Yisroel written by Rav Yisroel Najara, (the composer of Kah Ribon Olam from Friday night zemiros).  On the second day there is a minhag to read the Kesuba between Hashem and Am Yisroel.