We were asked about this. It's really hard to explain and I never do it well-- ask my students. Basically, it's about whether a noun is "restricted" by the modifying clause that follows.
To start, a modifier is a word or phrase or clause which tells us more about a noun or verb. (The RED dress... he said VICIOUSLY). Those that modify a noun are called "adjectival".
A clause is a sentence element that includes a noun or pronoun and its verb. The most common adjectival clause (modifying a noun) is the relative clause, which usually, not always, starts with a "wh" pronoun, particularly who and which.
The filmmaker Otto Filbenstein, who died in 1999, directed the remake of the French silent film C'est La Vie.
The writers who went to Hollywood after the war often achieved financial success beyond their wildest dreams.
See the difference there? The first has commas around the modifying clause, and the second doesn't.
The first is "non-restrictive" because it is just additional information about the noun. It doesn't "restrict" the noun to a smaller group. You could take that "who" clause out and you'd still have the main point-- that Otto remade that French film.
The second is "restrictive." That is, this refers only to those writers who went to Hollywood after the war. The noun isn't really just "writers," but "writers who went to Hollywood after the war". It "restricts" the noun to this smaller group. You can't take out the "who" clause in this one, because it's necessary to say who achieved financial success. (The writers often achieved financial success.... It's a legitimate sentence, but it isn't what you mean-- all writers don't often achieve this success-- you're talking about only the ones who went to Hollywood before the war.)
(Everyone should learn to diagram sentences, because it's much clearer in a diagram that "The writers who went to Hollywood after the war" is the noun phrase.)
Point is, Non-Restrictive= comma before and after, because it's 'unnecessary information," useful, perhaps, interesting perhaps, but inessential. You can pull it out and the sentence still means what you want it to mean.
Restrictive= no commas, because the clause actually is part of the noun.
I'll give some examples, but you all supply some too, and that'll help!
My brother Phil, who is two years older, ran for Senate and won. (NON, because his age is just additional information.)
Governors elected in the past two years often faced serious budgetary issues their first year in office. (RESTRICTIVE, because this refers only to those governors elected in the past two years. -- There's no "who" there, though it could be there--- Governors who were elected...)
N: Part-time students, who often have outside jobs, need to learn time management skills.
R: Students who haven't attended class the first week will be automatically dropped.
N.: The snow, which will be so grimy in a few days, is lovely tonight.
R: Coaches who choose to go for it rather than punt show trust in their offense.
N: A first-class upgrade, which allows the passenger more legroom, takes 60,000 points.
R: Bronco fans who move out of Denver can keep in touch through this Facebook page.
How about you all supplying some examples?