There is NO way to make this shorter. Just hang in there, go slow, and read it over.
To understand run-ons we need some definitions.
Definition of a sentence: subject, verb, complete thought.
Definition of a fragment: missing one of the above.
Definition of a run-on: too many of one of the above.
In other words most of the time a run-on is more than one sentence.
The most common run-on is the missing comma before a coordinating conjunction or the missing coordinating conjunction itself. But let's back up a bit.
There are only seven words that can connect two complete sentences. You can use the mimetic device fanboys to remember them: for, and, nor, but, or, yet, so.
The problem here is that student writers sometimes use however, therefore, then and many others to connect two sentences. Using these words that way would be incorrect.
However, the major issue with run-ons in student papers isn't that there is a missing period between two sentences or a missing comma before a coordinating conjunction or the missing coordinating conjunction itself.
The major issue with run-ons is a missing idea between two sentences.
Let's define the common coordinating conjunctions. Remember that these words are not always used as coordinating conjunctions and that we are only defining them as coordinating conjunctions.
and= also, with, etc.
So now we should realize that by leaving out a coordinating conjunction we have a missing idea between two sentences--also, however, therefore--which is usually the problem with run-ons. Let's take a look at a few.
He knew these words by heart, and he should have, this song was constantly playing on his computer.
There's more than one complete thought here, but let's see how to connect them.
The simple fix is a period: should have. This song
But we lose the idea between these two sentences. It sounds better like this:
and he should have because this song was
and he should have, for this song was
Incidentally, subordinate clauses that begin with words like because, like the one above, do not always have to comma. It depends on length, style and clarity.
Ha, but there is an exception to this correction. There is another solution. We could also do this:
and he should have--this song was
The dash is usually typed as two hyphens and usually shows an interruption of sorts and can be used for emphasis and style. By using it here it counts as an interruption, but somehow we also translate it as that missing idea between the two sentences. We translate it as for or because.
As for the general definition of a sentence, there is an exception as usual.
"I love you, I hate you, and I want my records back."
This is okay even though it is technically three complete sentences presented as a list with an and on the last one, exactly as if I had written,
I like beer, popcorn and movies.
It works with longer sentences, and I've seen it work many more than just three sentences.