Tefillin are prayer phylacteries worn on the arm and head of a pray-er during Shachrit, the morning service, although not on Shabbat.
There are two kinds of tefillin - tefillin shel yad (for the hand) and tefillin shel rosh (for the head). Each box contains extracts from the Torah which explain the mitzvah, the good deed, of performing the laying of tefillin. One of these extracts is the Shema.
Tefillin are strapped to the arm and head are recitation of a blessing. The tefillin shel yad will go on the opposite arm to the hand with which one writes - so that a left-handed person would wear his tefillin on his right arm, and vice-versa. The straps of this box will form a hebrew letter shin around the hand, and around the finger will form a dahlet. This is to form the three letters of one of the names of G-d (Shddai).
The binding of something onto a person is deeply symbolic. It is not only a spiritual binding to the prayer, but a physical one as well, which creates an extra potency.
If tefillin are not worn during Shachrit, and are put on during the day, it is a custom to recite the Shema. They may only be worn by those post bar/bat mitzvah age, and are not to be worn people in mourning before the funeral, or by a groom on his wedding day, or on a sabbath or festival.