The Tallit Katan, also casually referred to as Tzitzit, Tzitzis, or in Yiddish, Arba Kanfot (meaning four corners). is a four-cornered garment typically worn under the shirt, with some communities wearing it on top, with a fringe tied on at each corner of the garment. Technically, the fringes themselves are called Tzitziot (Tzitzit in plural), but the word can be used in casual conversation to refer to the entire garment.
The reason for this distinction is that the formal name for the garment is Tallit Katan, a small Tallit, or prayer shawl. But the word Tallit is used to refer to a larger garment, a shawl worn only during prayers. The Tallit also has Tzitzit strands (or fringes) but the two garments are referred to with different words out of convenience. Sephardi tradition prefers a woolen garment, in accordance with the Mishnah Berurah, Shulchan Aruch, and Rabbi Moshe Feinstein, but cotton is also commonly used and is an acceptable alternative worn by the Chazon Ish and the Vilna Gaon, Rabbi Joseph Soloveitchik, and most of German Jewry.
The Tallit Katan is worn every day by Jewish men and boys, with the strings at the corners reminding the wearer of his covenant with God.
The Bible verse responsible for this garment is Numbers 15:38-39 (בְּמִדְבַּר טו) which commands the Hebrews to
"make them throughout their generations fringes in the corners of their garments,"
or in Hebrew "וְעָשׂוּ לָהֶם צִיצִת עַל-כַּנְפֵי בִגְדֵיהֶם, לְדֹרֹתָם"
and then continues to detail the blue dye used on the fringes, which we no longer know for certain which dye was used. Some communities use a dye thought to be the correct one on some of their fringe, and some do not.
The verse continues with " וְהָיָה לָכֶם, לְצִיצִת, וּרְאִיתֶם אֹתוֹ וּזְכַרְתֶּם אֶת-כָּל-מִצְוֹת יְהוָה, וַעֲשִׂיתֶם אֹתָם" detailing how seeing the fringes all day will remind you of all of God's commandments so that you remember to do as God commands.
The requirements for a Tallit Katan and a Tallit Gadol (the prayer shawl) are identical, as they serve the same function and are extrapolated from the same verse. Thus, doing both isn't obligatory De'orayta (according to the scripture), but Rabbinically wearing a Tallit Katan is required and accepted custom in religious communities. The Tallit Katan is often worn from the age of Upsherin, 3 years old, when the child is capable of learning its meaning, unlike the Tallit Gadol which is worn only after Bar Mitzvah or marriage, depending on custom. Some Tallit Katan options still have the traditional strioes of the Tallit Gadol, but most do not.
Now, more options are available than just the customary rectangular piece of cotton or wool fabric with a neck hole cut out of the middle. Tzitzit now can involve a proper piece of clothing, such as a sleeveless or short sleeved shirt, still with four corners holding fringes, but closed at the sides to create a garment more easy to wear and possible to wear alone. In addition, options made in a mesh pattern exist, to allow a cool breeze to come through on a hot day or during exercise. These Tallit Katan options give the wearer more comfort and flexibility in daily life, while still fully fulfilling the commandment to wear the tied Tzitzit fringes. These new styles of Tallit Katan make a great gift for someone with a lifestyle that could benefit from them, as well as a terrific investment in your own comfort and daily worship.