Leave the commas out when the clause is telling the reader which one you mean, and put them in when the reader already knows which one and you are adding a fact about it. The students who had finished left early and The students, who had finished, left early are built from seven identical words and they are two different reports of the afternoon.
Without the commas, only some of the students left, and the clause is what picks them out. With the commas, all of them left, and the clause is an aside about the whole group. Nothing else in either sentence carries that difference, so the punctuation is not a matter of taste here in the way it is around an introductory phrase.
Eight sentences, and the comma that changes the meaning
- the man who lives next door. Who for people, which for things, that for either in this kind of clause.
- the book that I read. This clause identifies which book, so it takes no comma, and removing it would leave the sentence unclear.
- My brother, who lives in Cork, is a teacher. The commas make the clause extra information and imply you have one brother. Without them it implies several.
- the book I read. The pronoun can be dropped when it is the object of the clause, which is very common in speech and is not an error.
- the man I spoke to. The preposition stays at the end in ordinary English; moving it to the front is correct and formal.
- the house where I grew up. Where, when and why introduce clauses about place, time and reason without needing a preposition.
- the reason why is accepted, though the reason that and plain the reason are both cleaner.
- whose works for people and things alike: the car whose alarm went off is correct, if slightly formal.
The test that settles it in four seconds
Delete the clause and read what is left. The students left early, after deleting an identifying clause, no longer picks out the people you meant, so the clause was doing identifying work and takes no commas. After deleting an adding clause the sentence still means exactly what it meant, so the clause was an aside and takes commas on both sides.
Two smaller facts follow from that split rather than having to be memorised beside it. An adding clause cannot use that, only which or who. And the pronoun can be dropped only from an identifying clause and only when it is the object, which is why the book I read is ordinary English and my brother, I met yesterday, is a doctor is not a sentence at all.
Where it stops: a decade spent on that against which
Nearly every learner has been drilled on when to use that and when to use which, and almost none has been taught what the commas do. That is the stall, and it is unusual in being caused by teaching rather than by neglect. The that and which question is partly a house style preference that varies between publications and countries. The comma question is not a preference at all, and getting it wrong changes who left the building.
It survives because the cost is invisible from the inside. My brother who lives in Berlin is a doctor quietly tells your reader you have more than one brother, and they will believe it, and they will never mention it. You go on writing sentences that carry a fact you did not intend and nothing comes back to tell you.
Reading a hundred thousand words that obey the rule
The Guardian and Observer style guide is free to read online, in a searchable A to Z, and it is one of the few places where a publication's decision about defining and non-defining clauses is written down in public. It replaced the printed house style sheet, which existed at every newspaper and was seen only by the people who worked there. Read the entry, then read the paper, which is the useful half: the rule is applied consistently across everything published, so you get the explanation and then an unlimited supply of worked examples.
Know what a style guide is before you lean on it. Much of it records one publication's preference and is not binding on you. The commas around a non-defining clause are the exception, because they carry meaning rather than house style, and a guide that lists both kinds of entry side by side does not always tell you which sort you are reading.
What it looks like when it goes wrong
In writing the failure is a comma placed by ear, usually one comma rather than two: The students, who had finished left early. A single comma opens an aside and never closes it, and a reader hits the end of the sentence still waiting.
In speech the failure runs the other way. There is no comma to hear, only a pause and a drop in pitch, and learners routinely miss it and take an aside as an identification. The dropped pronoun makes it harder again: in the man I saw yesterday was late there are two verbs and nothing joining them, and a listener who is waiting for a joining word will still be waiting when the sentence ends.
Which slots a clause can occupy, and what may never come between a verb and its object, is where English insists, and where it lets you choose, and the punctuation half of this belongs with emails, essays and reports that read like a native wrote them, both under the rules that decide what a sentence is allowed to mean.
The question this opens is a reading problem rather than a writing one. If a relative pronoun can simply be absent, then a long sentence can contain a clause with no visible edges, and finding those edges at speed is a different skill from placing the commas yourself: getting through more of it without losing the meaning.