Use "that" with restrictive clauses. A restrictive clause is one that limits -- or restricts --the identity of the subject in some way. When writing a restrictive clause, introduce it with the word "that" and no comma. (However, if the subject is or was a human being, use "who" to introduce the clause.)
Example of correct Restrictive Use:
The painting that was hanging in the foyer was stolen.
Explanation: The use of "that" in this sentence is correct if the reader intends to single out the one painting that was in the foyer as the stolen painting. However, if there were several paintings hanging in the foyer, this use would be incorrect, since it would mislead the reader into believing that there had been only one painting in the foyer. The restriction here tells us that the one painting that had been hanging in the foyer was stolen -- not the painting in the living room, or the one in the drawing room, or any of those in the parlor.