Answer:
   
   The Torah commands us to wear tzitzit (fringes) at the corners of our
   garments as a reminder of the commandments [Num. 15:37-41, which is in
   the third paragraph of the Sh'ma, recited during the morning and
   evening prayers]:
   
     Speak to the Children of Israel and say to them that they shall
     make themselves tzitzit on the corners of their garments,
     throughout their generations and they shall place upon the tzitzit
     of each corner a thread of blue wool. These shall be your tzitzit,
     and when you see them, you shall remember all of God's commandments
     so as to keep them. You will then not stray after your heart and
     eyes after which have lead you to immorality. You will thus
     remember and keep all My commandments, and be holy to your God.
     
   This is reiterated in Dvarim (Deuteronomy) 22:12:
   
     You shall make for yourselves twisted threads on the four corners
     of your garment with which you cover yourself.
     
   This commandment only applies to four-cornered garments, which were
   common in biblical times but are not common anymore. 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"
   and is usually worn under the shirt. This garmet is similar to a
   poncho. The tallit katan is worn under the shirt, sometimes with the
   tzitzit hanging out so they can be seen.
   
   All garments of a certain size or larger that have at least four
   corners must tzitzit attached. The original requirement was to have a
   blue thread among the other threads. However, since the precise shade
   of blue is no longer known and the source of the dye used, only the
   other threads are used (except among certain chassidic groups that
   claim to know the dye formula). Typically, these threads are white.
   Why? Although technically, they can be of any color, there is a debate
   as to which color is the ideal: some say they should be white, some
   say the color of the garment. The question is avoided by wearing a
   whilte garment.
   
   Note: There is a complex procedure for tying the knots of the tzitzit,
   filled with religious and numerological significance. The tying
   pattern symbolizes the 613 traditional commandments in the Torah.
   
   Why do tallit typically have blue or black stripes? The reason why the
   tallis is striped is simply because that was the fashion in Greece and
   Rome. But this doesn't answer the question of why blue or black?
   Tzitzis are supposed to include a thread of blue wool in each tassle.
   Most believe we do not know the specific dye needed for the mitzvah.
   In memory of this dye, some adopted a custom to place a blue stripe on
   the garment itself. Others decided to add a black stripe of mourning
   for the lost element of the mitzvah. The black stripe gained
   popularity in Europe of the 15th through 19th centuries, when
   black-and-white clothing was more common for Jews in general. The blue
   stripe is now seeing a revival in the 20th and 21st centuries, but
   it's actually the older of the two customs. It just seems to us to be
   more modern. Sepharadic Jews believes the debate over what color is
   appropriate precludes wearing colored stripes, so they wear white
   stripes (or a different weave) on their talleisim. Maimonides was of
   the "same color as the garment" camp. For Baladi Yemenite Jewry (those
   Yemenite Jews that resisted the influx of Syrian customs), Maimonides
   is the final word on Jewish law. So, they do not wear a tallis of any
   particular color. One will often find an older, more traditional,
   Yemenite man wearing a rich blue or red tallis with matching strings.
   With or without stripes.
   
   A tallis can be made of any fabric. Ideally it should be wool or
   linen, as there is a rejected opinion that requires one of those two.
   However, since it's a rejected opinion, using anything else is no big
   deal. In practice, however, since you can't find linen strings to hang
   on the tallis and you can't put wool strings on a linen garment due to
   shaatnez, Wool is the norm (at least in Orthodox, Sepharadi, and
   Yemenite circles). Some even make a point of wearing a wool garment
   for the tzitzis worn under the shirt. As for the minority of the
   garment (if it is made of wool): assuming you avoid linen, any other
   thread can be included in the minority of the garment -- silk,
   artificial fibers or metal.
   
   During prayers, the custom is to wear a four-cornered shawl 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
   German-Ashkenazi communities, a boy will put on a tallis when he
   becomes a bar-mitzvah (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.