A restrictive clause is sometimes called a defining relative clause, because it defines, or somehow identifies, a person or an object. For example:
The cat which belongs to Mr Jones is ginger. (That is, not the cat which belongs to Mrs Smith.)
In this sentence, if you remove the restrictive clause, it is not clear which cat you are talking about.
A non-restrictive clause is sometimes called a non-defining clause, because it doesn't actually define; instead, it simply provides extra information:
I know the cat you are talking about. The cat, which belongs to Mr Jones, is ginger.
In this case, we can leave out "which belongs to Mr Jones" and we can still understand the sentence: we still know which cat is ginger. The fact that it belongs to Mr Jones is extra information.
Because it's extra information, we use commas. Or we could use parentheses:
The cat (which belongs to Mr Jones) is ginger.
In your example, the first sentence. "My wife who lives in Mumbai is a doctor," means that you have more than wife -- but it is the wife who lives in Mumbai (not the one who lives in Kolkata) who is a doctor.
With the commas, the sentence means that you have only one wife: she is a doctor, and she lives in Mumbai.
- For Teachers