Soul Gems are a holding place for a magical girl's soul and the reason they can transform into their battle gear.
Kyubey explains that the body is just a container possessed by a soul. Everyone was shocked and angry to hear this, claiming it made them into "zombies", but to Kyubey this is just simple logic, since he has no understanding of human emotions. Because their life source is placed in these containers, if taken from a girl they will "die". This happened to Sayaka Miki in episode 6 when Madoka Kaname throws it, but Homura Akemi soon returns it.
A magical girl must use grief seeds to cleanse their soul gem, so they don't turn into a witch. When a soul gem is not cleaned, the girl's soul gem is tainted and breaks, opening the girl's witch labyrinth.