A run-on is two sentences together as if they were one: I went to bed I was tired.  That has two subjects and two verbs, and neither clause is dependent.  In such a simple example, the fix is simple:  I went to bed.  I was tired.  OR I went to bed because I was tired.  OR  I was tired, so I went to bed.

The "because" clause is dependent now on the main clause.  If you use it by itself, with no independent clause, it is considered a fragment:  "Because I was tired" is a fragment.  Also, "which you gave me," "that I told you about," and other dependent clauses.  The last thing I wrote is a fragment. To fix it, I need to add a verb:  "Which you gave me" "that I told you about," and other dependent clauses are fragments.

Run-on sentences are two sentences that should properly have been joined by a conjunction, but weren't. 

I read the book I liked it a lot.

That's a run-on sentence. 

I read the book, and I liked it a lot. 

That's how you fix a run-on.

Sentence fragments? That's what you call a sentence that has an incomplete thought. 

When I go to the library. 

That's a fragment. 

When I go to the library, I look at the newest books. 

That's how you fix it - by finishing the thought.

fragment : Either the subject or the predicate is missing.

Mary decided to ...........

went to the Fair( subject missing)

..................

A run on is several sentences combined with a conjunction or proper punctuation.

Examples of run on and a correct one.:

The boy went to the show he saw a friend it was great

The boy went to the show; he saw a friend, and it was great.