If you join two independent clauses (a subject and verb that expresses a complete thought aka, a complete sentence), you need a comma and a conjunction.

Conjunctions are the words that link words and phrases. They are also known as the "FANBOYS"

For, and, nor, but, or, yet, so

(Ex.) The sentence I ran up the hill and the sentence Jill ran down the hill are both independent clauses. They could be joined in the following way:

I ran up the hill, but Jill ran down the hill . (Correct)

OR

I ran up the hill, and Jill ran down the hill. (Correct)

Be sure to use conjunctions that make logical sense. The conjunction nor would not work in this example.

If you join the main clauses with only a comma and leave out the conjunction, that is a comma splice.

(Ex.) John went to the mall, he bought a pretty shirt . (Incorrect: comma splice)

John went to the mall , and he bought a pretty shirt. (Correct)

In this example, each clause can stand alone as a complete sentence, so the conjunction is necessary to put them together.

Be wary of getting "comma happy." You do not need a comma before every conjunction, only when it is joining two independent clauses.

(Ex.) John bought a pretty shirt , and a hat. (Incorrect)

John bought a pretty shirt and a hat . (Correct)

In this example "and a hat" could not stand alone, so it does not need a comma.

You need to separate the items with commas when you are listing a series of items or phrases.

(Ex.) I had green beans macaroni and cheese and chicken for dinner last night . (Incorrect)

I had green beans , macaroni and cheese , and chicken for dinner last night. (Correct)

There is debate over whether or not to include a comma in front of the conjunction that comes before the final item in the series (and, nor, or). Whether or not you choose to insert that last comma, it is important to be consistent.

(Ex.) Sarah swam ten laps in the pool , biked over the mountain , and ran five miles. (Correct)

Sarah swam ten laps in the pool , biked over the mountain and ran five miles. (Correct)

When you use two adjectives that modify the same noun, they are coordinate adjectives and you must separate them with a comma.

(Ex.) I like to drink strong sugary coffee . (Incorrect)

I like to drink strong , sugary coffee . (Correct)

And

I have a bright, red mug . (Incorrect)

I have a bright red mug . (Correct)

In the first example strong and sugary both modify coffee, so a comma is needed to separate them. In the second example, bright is an adjective that modifies red , so a comma is NOT needed.