Two independent clauses (each has a subject and a verb and makes sense when read alone):

My dog went to the park.              I threw his ball.

Do not join only with a comma--the dreaded comma splice:

My dog went to the park, I threw his ball.

Add all the adjectives and prepositional phrases you want--no commas (generally):

My big lovable dog went to the park in a red wagon with white stripes.

Double the subjects or the verbs, making no changes:

Fred and I threw his ball.

I threw his ball and a stick.

My dog went to the park and rolled in some smelly stuff.

Connect with semi-colon or with comma plus coordinating conjunction ("and," "or," "but," "for," etc.):

My dog went to the part; I threw his ball.

My dog went to the part, and I threw his ball.

Subordinate the first clause and use the comma:

When my dog went to the park, I threw his ball.

After my dog went to the park, I threw his ball.

Because my dog  went to the park, I threw his ball.

Subordinate the second clause and (generally) do not use the comma:

My dog went to the park because I threw his ball.

My dog went to the park after I threw his ball.

My dog went to the park, while I threw his ball.
 
 

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