Identify essential phrase

A phrase in a sentence is either essential or non-essential, and telling which is the key to punctuating it.
An essential phrase gives information the sentence needs to be clear.
A non-essential phrase only adds extra detail the sentence can do without.

A non-essential phrase is wrapped in commas — one comma before it and one comma after it.
The way to check is simple: read the sentence without the phrase, and see if it still makes sense and still points to the same thing.
If it does, the phrase is non-essential.

Take the sentence: "My oldest brother, who lives in Chicago, is a doctor."
First read it without the phrase: "My oldest brother is a doctor."
That still makes complete sense, and you still know exactly who is meant.
Now read it with the phrase added back: "My oldest brother, who lives in Chicago, is a doctor."
The phrase just tells you a little more about him, so it is non-essential and belongs inside commas.

An essential phrase is different, because the sentence needs it.
Take the sentence: "The student who scored the highest won the prize."
Read it without the phrase: "The student won the prize."
Now you cannot tell which student is meant, so the meaning is lost.
Because "who scored the highest" is needed to point to the right student, it is essential and takes no commas.

Is the phrase "a small town in Italy" essential or non-essential in this sentence: "Last year we visited Verona, a small town in Italy, during our trip"?
Read it without the phrase: "Last year we visited Verona during our trip."
That still makes sense and still names the place, Verona.
So the phrase only adds extra detail — it is non-essential, which is why it sits between commas.

Is the phrase "wearing the red jersey" essential or non-essential in this sentence: "The player wearing the red jersey scored the winning goal"?
Read it without the phrase: "The player scored the winning goal."
Now you cannot tell which player is meant.
Because the phrase is needed to point to the right player, it is essential and takes no commas.

Is the phrase "our family dog" essential or non-essential in this sentence: "Max, our family dog, loves chasing the ball"?
Read it without the phrase: "Max loves chasing the ball."
That still makes sense and still points to Max.
So "our family dog" is extra detail — it is non-essential and belongs between commas.