Run-on sentences are grammatically incorrect; they read very badly and should be rooted out from your writing. Unfortunately computer grammar checkers are not good at identifying them. Grammar checkers will identify the comma splice (two independent clauses separated by a comma) but seem quite happy if there is no punctuation between the clauses.
A run-on sentence is really two sentences that should be separated by some kind of punctuation mark but are not.
Below are some examples of run-on sentences. Although they look very clumsy when seen in isolation like this, it is easy to get carried away when writing an essay and end up with run-on sentences.
She only rings me at certain times she just wants to make herself feel better.
I rushed out to the shop I had no milk left.
The professor ran to his office he had just had a brilliant idea.
If you find that your sentences are long and each contains more than one idea, you will need to find a way of separating the ideas. You could simply use a full stop. A semi-colon might make the connection between the ideas clearer. You could connect the two clauses with a conjunction of some kind.
I rushed out to the shop. I had no milk left.
I rushed out to the shop; I had no milk left. (The connection between the two clauses is implied but not explicitly stated.)
I rushed out to the shop because I had no milk left.
I rushed out to the shop, as I had no milk left.
There is no substitute for proof reading your writing to eliminate run on sentences.
Test your ability to spot run-on sentences with this exercise.