Sentence fragments are basically incomplete sentences. The may consist of a predicate without a subject. Or a subject without a predicate along with incorrect punctuations. Here's an example below.
Just like her!The sentence doesn't include an actual verb. Like it doesn't really say what is exactly just like her.It also has very incorrect punctuation. It should have ended with a period not an explanation point.Being that explanation points express emotions but the fragment doesn't include any expression at all.
Run-ons
Run-on sentences are compound sentences that aren't punctuated incorrectly. They are supposed to include either commas,semiclons or conjunctions with semicons to indicate a break. Here's an example below.
I like dogs she likes cats.The sentence should have included but and a comma to express both thoughts. Another example could have been the following.We went to the mall although he went to the movies.The sentence should have included a comma after although because it's supposed to be a compound sentence.