A tallit is a prayer shawl. It may be pronounced “ta-LEET” or “TA-lis” depending on the kind of Hebrew spoken. (The plurals, respectively, are “ta-lee-TOTE” and “ta-LAY-seem.”) The shawl itself is just a shawl; the important parts of the tallit are the long knotted fringes or tzitzit (tzeet-TZEET). We wear them to remind us of the 613 mitzvot [commandments].
Jews wear a tallit for morning prayers. The person who leads prayers often wears a tallit no matter what time of day. We get the commandment to wear the tallit from two places in the Torah: Numbers 15:37-40 and Deuteronomy 22:12. You can learn more about the meaning and history of the tallit from this article by Rabbi Louis Jacobs.
A tallit is one of those things reserved for people who were born Jewish or who have been through the process of conversion. The purpose of the tallit is to remind us of our 613 sacred duties (mitzvot). Only a person who is bound by those duties needs to be reminded of them.
Occasionally you may see a tallit with blue cords in the fringes. Blue is a difficult dye to find in nature. In ancient times, Jews fulfilled the direction for a blue cord by using something called techelet, a product from sea snails, knowledge of which was lost in the Middle Ages. Recently, scholars have come to believe that techelet is a dye made from the murex, a sea snail, so some Jews have begun wearing techelet fringes again.