It means vomiting. The porcelain god is your toilet, and you have to kneel before your toilet to vomit. The phrase is usually used after heavy drinking. "She was so drunk, she ended up praying to the porcelain god." (she was was so drunk she ended up vomiting.)

Just to give another expression or two... we also use "drive the porcelain bus" and "talk on the big white phone (to God)". XD