A run-on sentence is actually two or more sentences shoved into the same space. The two most common varieties of run-on sentences are the fused sentence, and the comma splice.
A fused sentence is easy to spot, as it doesn't make much sense the way it is written. A good example of a fused sentence is, "We love the beach we go every summer." This is made of two independent clauses strung together without benefit of punctuation. (A clause is an idea. An independent clause is an idea which is complete and can stand alone.) The clause "we love the beach", and the clause "we go every summer" are actually two separate ideas. The way to correct a fused sentence is to separate them with a period and capitalize the second sentence. Our fused sentence should read, "We love the beach. We go every summer."
A comma splice is similar to a fused sentence, except it has a comma placed in the middle of it. "The milk had gone sour, I threw it out." is an example of a comma splice. Independent clauses ought to be separated by a period rather than a comma. This sort of run-on sentence is a lot harder to recognize, since the comma between the clauses simulates a natural speaking pause. Most comma-spliced sentences are quite long; this can make them difficult to read.
As a general rule of thumb, sentences in works of fiction should average thirteen words. This does not mean you should make every sentence thirteen words long. If all your sentences are exactly the same length, the prose becomes stilted and awkward. You need sentences of all lengths, both longer and shorter, but over the course of your manuscript, they should average near thirteen words each. If your average is a lot higher than thirteen words, start looking for fuses and comma splices that can be broken in half before you go rewriting some of your more elegant prose.