Run-on sentences are independent clauses that have not been joined correctly. If two independent clauses appear in one sentence, they must be joined either with a comma and coordinating conjunction or with a semicolon (and on rare occasions with a colon or dash). You could also choose to make the independent clauses into two separate sentences, or make one of the clauses dependent on the other by adding a subordinating conjunction.
Most run-on sentences are comma splices; that is, independent clauses joined by a comma but lacking the coordinating conjunction:
Willie Nelson has been showing strong Mexican influences, his work has improved.
Inserting the coordinating conjunction "and" after the comma corrects the comma splice:
Willie Nelson has been showing strong Mexican influences, and his work has improved.
Or you could subordinate one of the clauses:
Since Willie Nelson has been showing strong Mexican influences, his work has improved.
Or, as we mentioned above, you could replace the comma after "influences" with a semicolon or a period (with appropriate capitalization afterward, of course).
Another form of run-on sentence, the fused sentence, occurs when you neglect to put any punctuation mark and/or coordinating conjunction between two independent clauses:
My mother was in a good mood I took the opportunity to ask for money.
Insert a period (or semicolon) after "mood," or use an appropriate conjunction and connect the clauses with a comma:
My mother was in a good mood, so I took the opportunity to ask for money.
Adding a comma and the coordinating conjunction "so" properly joins these two independent clauses.
Because my mother was in a good mood, I took the opportunity to ask for money.
Adding the subordinating conjunction "because" makes the first clause dependent on the second.
To learn more about writing sentences that don't run on and on, check out the Elements of Effective Writing II: Form and Composition.