Deliberate use of sentence fragments: Sometimes writers deliberately use fragments to create emphasis. For instance, "Any student facing three exams in one day has but one thought. Help!" In The Eternal Summer, a book about golf in 1960, Curt Sampson artfully uses sentence fragments to mimic Ben Hogan's putting difficulties. Hogan had become extremely anxious about his putting, and Sampson uses fragments to suggest that Hogan's once fluid putting stroke had become stiff and mechanical. Sampson writes, "Hogan just couldn't. Start. The clubhead. Back." If you chose to use a fragment, make sure that the surrounding context highlights your decision as deliberate.
Run-ons are independent clauses that have either not been separated at all or have been separated by a comma alone. The comma that separates these independent clauses is called a comma splice. There are several ways of marking off independent clauses so that the reader knows where these units begin and end. Among these choices are periods; semi-colons (the semi-colon, which might well be called a semi-period, says that the units marked off are very closely related); colons; dashes; and the comma plus conjunction such as but, and, or or.
Long sentences aren't necessarily run-ons and writers shouldn't fear writing longer sentences because they think that longer units will mysteriously turn into run-ons. "I laughed, I cried" (two subject + verb+ completed thought units) is a run-on. As with fragments, run-ons can be deliberately used for effect, but unless you make your intentions clear, avoid them.
The best way--by far--to recognize what is and isn't a sentence is to read your writing aloud. It is much harder to fool your ear than your eye.