Comma Rule: Use a Comma When Introducing an Independent Clause with a Conjunction

Independent clauses work in the same way as full sentences, so you normally join two independent clauses with a semicolon. If there is a conjunction linking the independent clause, then you use a comma.

Correct
I am jealous of Danthony’s shoes, but I will not steal them.
I have very little impulse control, and now I have a new pair of shoes.

The independent clause is linked by a conjunction and a comma. You may also replace the comma and conjunction with either a semicolon or a period to make it two complete sentences. 

Oh, Danthony, I couldn’t resist; your sense of style is just too cool and I was driven to crime.

Incorrect
I am jealous of Danthony’s shoes, I will not steal them.
I have very little impulse control, now I have a new pair of shoes.

These examples are incorrect because two complete sentences cannot be linked with only a comma.