The answer to this question really varies depending on your ancestry and/or community custom. Allow me to share some information:
Jewish men and boys traditionally wear a small fringed tallit (lit.: cloak) under their shirts throughout the day. This garment is known as a tallit kattan (a "small" tallit), and even little children – as young as three years of age – are taught to wear this light four-cornered garment.
The tallit worn during prayer is much larger, it is known as a tallit gadol ("large" tallit).
Traditionally, Eastern European Jewish men begin wearing a tallit gadol for the daily prayers after marriage. Customarily the tallit is a gift from the bride or her family.
Among Western European and Oriental Jews, boys begin wearing the tallit gadol from the age of Bar Mitzvah or even earlier.
Click here for more on the tallit gadol.