A comma splice comes about when two independent clauses are connected together with a comma rather than something with a bit more bite.
This is something seen quite often in texts from inexperienced writers, who are perhaps working on the basis that commas mark pauses similar to those in speech and maybe do not know as much about grammar as other, more pedantry-hardened veterans.
An example of comma splice
He left the house to get some petrol, he drove all the way to Manchester.
This would probably get your grammar checker flagging the comma in between the clauses in green, but the problem can be remedied in several ways.
He left the house to get some petrol, and he drove all the way to Manchester. (Note the presence of the comma between the clauses.)
After he left the house to get some petrol, he drove all the way to Manchester. (Another useful comma crops up here too to separate the newly dependent first clause from the second.)
He left the house to get some petrol. He drove all the way to Manchester.
Or, perhaps the closest to using the comma,
He left the house to get some petrol; he drove all the way to Manchester.
The semi-colon still allows the two sentences to share a connection while still recognising the grammatical fact they are independent clauses.
The comma splice is rarely considered correct in English, although there are plenty of examples of it passing unchallenged in fiction and especially in poetry. Julius Caesar is usually translated into English as writing “I came, I saw, I conquered”, and not many people profited from an argument with him.
In some other languages, like Bulgarian, it is actually compulsory. This just underlines how tricky the comma is and just how powerful such a humble little mark can be.