Rule: WHICH is used with a comma to introduce a non-essential clause (can be left out without changing the meaning of a sentence).
- 
Correct:  Brush your teeth with toothpaste containing abrasives, which constitute about 50 percent of typical toothpaste.
 - Incorrect: Brush your teeth with toothpaste containing abrasives that constitute about 50 percent of typical toothpaste.
 
Rule: THAT is used without a comma to introduce an essential clause (one that is essential to the meaning of a sentence). 
- 
Correct:  Dogs that are hypoallergenic tend to shed and affect allergy sufferers less.
 - Incorrect: Dogs, which are hypoallergenic, tend to shed and affect allergy sufferers less.
 
Rule: THAT is used as a conjunction to introduce a dependent clause when a sentence sounds or looks awkward without it.
- 
Example: The woman said Tuesday that she had left her purse at home.
 
Rule: After some verbs, THAT is usually required:
- 
assert
 - 
contend
 - 
declare
 - 
estimate
 - 
make clear
 - 
point out
 - 
propose
 - 
state