The Nefillat Apayim prayer is one that expresses sorrow concerning our errors and shortcomings, which prevent us from standing before Hashem. Instead, we fall on our faces. For that reason, it is not proper to recite it on days of joy. Since, in its essence, the Tachanun prayer is voluntary, it is therefore customary not to recite Tachanun even on days that are deemed uncertain as to whether or not they should be defined as days of joy (the sources for these laws can be found in Shulchan Aruch 131:6-7 and in the commentaries).
On the following days, Tachanun is not recited: Shabbat, Festivals, Chol HaMo’ed, and Rosh Chodesh. The remaining days are presented in the order of the calendar year: Shacharit of Erev Rosh HaShanah; Shacharit of Erev Yom Kippur; from the 11th of Tishrei until the end of the month of Tishrei, since those days surround the Festival of Sukkot; Chanukah; Tu BiShvat; the two days of Purim, and in leap years also on the 14th and 15th of Adar I; the entire month of Nisan, since most of it possesses sanctity, for the dedication of the Tabernacle was held in the beginning of the month and the holiday of Pesach follows; the 14th of Iyar, which is Pesach Sheni; Lag BaOmer; from Rosh Chodesh Sivan to the 12th of Sivan, since the supplementary sacrifices for Shavuot were offered until that time; Tish’ah B’Av (either because of mourning or because it is called a mo’ed); and Tu B’Av. Additionally, at the Minchah service preceding all these days, Tachanun is not recited, except for Minchah of Erev Rosh HaShanah and Erev Yom Kippur, when many people recite Vidui then, but not Tachanun.
Today, ever since Hashem began to bring forth our redemption, Tachanun is not recited on Yom HaAtzma’ut (Israel’s Independence Day), and on the 28th of Iyar, Yom Yerushalayim (Jerusalem Liberation Day), nor at Minchah before each of them.