Steven's gem is called Rose Quartz. Steven has a gem because his
mother Rose Quartz gave it to him. Steven is without a mother
because Rose gave up her physical form to have a son. This is
picked up as the show gets older. Though Steven does not have a
mother, he has a father. He does not live with Greg Universe
(father) because he could not support Steven. He did not have the
money.
The Crystal Gems (Garnet, Amethyst and Pearl) raise Steven as
one of their own, since they all have gem weapons. Rose and The
Gems' were all friends, so Rose thought it would be a good idea for
them to house her unborn son. Throughout the show, the Gems' teach
Steven how to become one of them, and how to weild his special
weapon; his shield.