A "comma splice" happens when two complete sentences are "spliced" together with a comma, much like wires can be spliced together.
Consider:
I didn't need to tell her, she already knew.
There are actually two complete sentences here: I didn't need to tell her can stand on its own as a complete sentence. She already knew can also stand on its own as a complete sentence. Because we have two complete sentences stuck together with a comma, it's a run-on sentence and thus not proper grammar.
How do we fix it? One way would be to just swap the comma for a period: I didn't need to tell her. She already knew. This is proper grammar, but such short sentences don't look good in formal writing. We could make the sentence legal and interesting by saying I didn't need to tell her because she already knew.
Let's look at another one:
I made the changes, however, I'm not sure if the paper is ready yet.
Again, there are two complete sentences here: I made the changes is its own sentence. I'm not sure if the paper is ready yet is also a complete sentence. Here we have two complete sentences stuck together with "however" and a comma. This is not acceptable grammatically. We could go with Although I made the changes, I'm not sure if the paper is ready. In this case, "Although I made the changes" cannot stand alone as a sentence, so we don't have two complete sentences and thus we don't have a run-on. We could also go with I made the changes, but I'm not sure if the paper is ready yet. In this case "but I'm not sure if the paper is ready yet" is not a complete sentence, so, again, no problem.
How about one more:
Another example is Ecuador, this country also exports petroleum.
Once again, we have two full sentences stuck together with a comma. We could fix this by just rewriting as Ecuador is another country that exports petroleum or simply Ecuador also exports petroleum.
Now take a look at the examples below and see if you can identify whether the sentence is a comma-splice run-on or whether it's acceptable. If it's a comma-splice run-on, see if you can think of a good way to fix it. Click the "down arrow" to see if you're right.
Comma splice. One better option would be "I think they're smart because they always know the answers."
Not a run-on. It's fine, because "so we'll be fine" cannot stand alone as a sentence. We do not have two complete sentences.
Run-on comma splice. Both halves of the sentence can stand on their own, so it's a run-on. One better option would be "Although the first anagram was solvable, the second was designed to be impossible to solve."
Run-on comma splice. Both halves of the sentence can stand on their own, so it's a run-on. How about "Although there is much research on this topic, none has yet tested this effect among children."