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 it this is just simple logic, since it has no understanding of human emotions. Because their lifeform is place in these containers, if taken from a girl they will "die". This happened to Sayaka in episode 6 when Madoka Kaname throws it, but Homura 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.