Meaning of "My friend, who lives in Paris, is a teacher" with and without commas
In the first, who lives in Paris is an integrated relative clause. In the second, it’s a supplementary relative clause. Those, at least, are the terms used in ‘The Cambridge Grammar of the English Language’. Such clauses are also known as ‘defining / non-defining’ and ‘restrictive / non-restrictive’.
The first sentence suggests that the speaker has more than one friend. It allows us to believe that the speaker may have friends who live in New York, London and Berlin, as well as the one who lives in Paris. From the second, we understand that the speaker has only one friend and that that friend lives in Paris. It is supplementary, because we can remove the clause and still be left with a viable sentence. A supplementary relative clause merely provides additional information.
In writing, it is the convention to place commas around a supplementary relative clause, as shown. It is possible to indicate more clearly when a relative clause is integrated, rather than supplementary, by substituting that for who (or which). A supplementary relative clause is not normally introduced by that. In speech, a supplementary relative clause is indicated by intonation and by a slight pause at the start and end of the relative clause.
EDIT:
OK, OK. Sentence 2 envisages only one friend in the context of the sentence. Here’s how the clauses might occur for real.
Sentence 1
I know people in several European cities, and I suppose they’re all my friends, really. The one in Berlin, I don’t know her quite so well as the others, but my friend who lives in Paris, now she’s really great fun.
Sentence 2
So, this friend of mine told me what happened to her last year. I was really amazed, but tell me what you think. See, my friend, who lives in Paris, was on her way to work one morning when she saw what she thought was an accident in the Rue de Rivoli . . .
My friend who lives in Paris is a teacher.
"I am going to tell you something now about my friend who lives in Paris. I am telling you nothing about my other friends. This friend is a teacher."
"...who lives in Paris..." is an integral part of the statement to identify the friend in question.
My friend, who lives in Paris, is a teacher.
"My friend is a teacher. By the way, they live in Paris".
"..., who lives in Paris, ..." is an addition to the statement.
Pretty much the only semantic difference here, is that the first suggests that they have only one friend who lives in Paris, while the second adds in their living in Paris as extra information.
However, even this is not emphatically stated. If you were already talking about e.g. Paris accommodation prices, or the education system in France, then you might say "My friend who lives in Paris is a teacher. They found it hard to find somewhere nice to live on a teacher's salary." Or "My friend who lives in Paris is a teacher. She's got plenty of complaints about how they do things, but some of their ideas seem better than ours".
In those cases, if you happen to also have a friend who lives in Paris, and who is a wealthy theatrical agent, you just aren't mentioning them because they're irrelevant to the context already given.
The first sentence tells us which friend you are discussing and implies that you have more than one. The phrase who lives in Paris is the criteria to identify which friend.
In the second sentence, who lives in Paris is an aside, an incidental piece of information, often called a paranthetical clause. It does not identify which friend, it just tells us something secondary about the friend you are describing. It also gives no indication about whehter or not you have more than one friend.
The first is a more integrated sentence and emphasizes both on the fact of living in Paris and being a teacher. The second emphasizes more on the fact that your freind is a teacher. The part between the commas is an additional explanatory clause. You can drop the clause altogether with no loss in meaning– at least that's how it would be taken in written English.