Theory:

The Sentence 5 - The Function Of Commas In Sentences

Except for simple sentences, every sentence is a combination of 2, 3 and sometimes 4 different ideas. The function of commas is to distinguish between these ideas, so that the reader can understand the relationship they have with each other.

That is why there is a comma before the conjunction in a compound sentence, because that is where the two ideas meet.
- He sold his car, but he didn't sell his bike.

Very often extra information is given to support an idea in a sentence, and commas are used to indicate where this extra information starts and finishes.
- My brother, who is a doctor, lives in London. ('who is a doctor' is extra information)

So they work as separators.

They separate a phrase lacking a verb from the remainder of the sentence.
- "Having said that, I still have my doubts."

If they are over used, or do not clearly identify the different ideas because there are too many of them in a sentence, then their value is lost.

There should be just the minimum number needed to do the job.

Finally, A comma indicates where two ideas are joined, it doesn't join them. So, a comma without a conjunction is a mistake.
- He rang the bell, the door didn't open. IS INCORRECT.
- He rang the bell, BUT the door didn't open. IS CORRECT.