Thanks, Wikipedia.

 An English non-restrictive relative clause is preceded by a pause in speech or a comma in writing, whereas a restrictive clause normally is not. Compare the following sentences, which have two quite different meanings, and correspondingly two clearly distinguished intonation patterns, depending on whether the commas are inserted:

(1) The builder, who erects very fine houses, will make a large profit. (non-restrictive)
(2) The builder who erects very fine houses will make a large profit. (restrictive)
(1) The building company, which erects very fine houses, will make a large profit. (non-restrictive)
(2) The building company that|which erects very fine houses will make a large profit. (restrictive)

Of the two, only which is at all common in non-restrictive clauses.

(apparently Americans favor using “that” where possible - a partial victory for the prescriptivists?)

“When a comma can be inserted, the word is which.”[6] A simple test is to consider whether the clause is essential to the meaning of the sentence and whether removing it significantly changes the meaning of the sentence; if so, usethat. For example:

(1) The pitch that changed the outcome of the game came in the eighth inning.
(2) The fateful pitch, which came on a 2-1 pitch, struck the batter.

and of course,

Jack built the house that I was born in.
Jack built the house I was born in.

(“zero relative pronoun”)

(“that” is also a determiner, e.g. “That dog died.”)