WJamie and Cersei are a couple of twins who have carried on a seriously disturbing incestuous relationship for years, and their father either didn't know or never acknowledged it. This relationship results in three kids who should clearly not be sitting on the Iron Throne.
The theory:
In the books, it's stated that the Mad King Aerys took some liberties with Tywin Lannister's wife, Joanna, on Tywin's wedding night. Also, as Reddit user tuna_HP points out, there are hints in the books that seem to support the idea that Cersei and Jaime might not be Tywin's. This can be seen in a statement said to Jaime by his aunt:
Tyrion is Tywin's son, not you. I said so once to your father's face, and he would not speak to me for half a year. Men are such thundering great fools. Even the sort who come along once in a thousand years.
This, along with the fact that incest was a pretty common thing with the Targaryens, has led many to believe that perhaps Jaime and Cersei aren't Lannisters after all. If that's true, Cersei's children might actually have a claim on the kingdom, and Jaime would be a kinslayer, not just a Kingslayer.