The day after the festival is called Isru Chag (lit. "Attend the offering"). This name is derived from a verse (Tehillim 118:27). In addition, the day after Shavuot is called Yom Tvoach, the Day of Offering.
The other pilgrimage festivals (Passover and Sukkot) have intervening days when people, in Temple times, offered festive offerings in gratitude to God for having the opportunity to spend another festival in Jerusalem with the rest of the Jewish people. However, Shavuot lasted only one day, so these offerings had to be made on the day after the festival (actually, there was time until the 12th of Sivan to bring these offerings). This is why supplications (tachanun) are not recited in prayer on the days following Shavuot (up to and including the 12th of Sivan).