To become a vampire a human must be almost completely drained of his blood. He is dying. Then he must drink a substantial amount of vampire blood and he will become a vampire himself.
Sookie drinks a lot of Bill’s blood, but she isn’t drained of her own blood, that’s why she doesn’t become a vampire.
Update 17 november
Episode 11 provides a better explanation of how a vampire is “made”. Not only does the victim have to be drained by a vampire, then given the vamp blood, but they have to spend the night with the maker in a hole all buried up in a grave. That vampire’s magic essence is transferred to the victim, who sleeps two more days without the maker then awakens vamp.