Comma Splices
A comma splice is inserting a comma between two independent clauses without a conjunction.
Incorrect: I like Warren, he is very funny. (commas alone cannot join independent clauses without a conjunction)

Correct: I like Warren. He is very funny.
               I like Warren; he is very funny.
               I like Warren because he is very funny.

 
Fused/Run-On Sentences
A fused sentence is two or more independent clauses not separated by punctuation. This error is also known as a run-on sentence.
Incorrect: My sister has a lot of experience I've learned a lot from her. (two independent clauses are connected without any connector)

Correct: My sister has a lot of experience. I've learned a lot from her.
               My sister has a lot of experience; I've learned a lot from her.
               My sister has a lot of experience, and I've learned a lot from her.

 
Sentence Fragments
Sentence fragments are dependent clauses or phrases used as sentences.
Incorrect: While I was talking to BJ about what Jacob said to her. (this is a dependent clause which cannot stand alone)

Correct: While I was talking to BJ about what Jacob said to her, Jacob snuck up behind us, scaring us half to death.