In the Torah, there is a commandment to wear "fringes" on the corners of garments. That is, all garments of a certain size or larger, which have at least four corners, must have strings known as tzitzit attached. The original requirement was to have a blue thread among the white threads; however, since the precise shade of blue is no longer known and the source of the dye used, only the white threads are used (except among certain chasidic groups that claim to know the dye formula).
Since the normal clothing in our time does not have four square corners, traditional Jews wear a garment that is specifically made to have four corners so that the mitzvah can be fulfilled. This is known as the tallit katan or tzitzit and is usually worn under the shirt. Some people wear them with the tzitzit showing, others conceal them. The verses giving this commandment are found in the third paragraph of the Shema, which is recited during the morning and evening prayers.
During prayers, the custom is to wear a large rectangular garment with tzitzit (tallis gadol) and pray while wrapped in it. There are different customs as to when this is done. Most Ashkenazic men will begin wearing the tallis when they get married. In some Sephardic and GermanAshkenazi communities, a boy will put on a tallis when he becomes a barmitzvah (13 years old). There are some communities that begin this earlier. Customs vary among liberal Jews as to who wears a tallis and when it's worn.