Relative clauses are tricky, and most English speakers do not know the rules for which relative pronoun to use. This will be an over-simplified explanation.
1. If the information in the relative clause is essential to the meaning of the sentence, then use that with no punctuation.
The plane that crashed is being fixed.
2. If the information in the relative clause is not essential, then use which and enclose the clause in commas.
The plane, which has red leather seats, is being fixed.
In sentence 1 we know that the plane is being fixed because it crashed.
In sentence 2 we don't know why the plane is being fixed, but it isn't because of the red leather seats (this is just extra information).
In British English, we can use that/which or that/who when the clause is essential pretty much interchangeably, though 'that' tends to be used less in formal writing, so our position seems to be the same as Riverkid's Canadian English.