Imagine you have a castle with a big wall around it. Inside the wall there are two parts: the Motte and the Bailey. The Motte is like the top of a hill where the king and the important people live. It's small but very defensible. The Bailey is like the area around the hill, where the regular people live and work. It's much bigger but less defensible.
Now let's say someone wants to attack the castle. It's very hard to attack the Motte because it's up high and has great defenses. So instead, the attacker focuses on the Bailey. The Bailey is much easier to attack because it's big and the walls aren't as strong. The attacker starts to destroy the Bailey and soon the people living there are in big trouble.
The king sees this happening and decides to do something. He can't defend the entire Bailey because it's too big, but he can defend the Motte. So he retreats to the Motte with all of his knights and army. From the Motte they can shoot arrows and throw rocks down at the attackers, who are stuck trying to destroy the Bailey.
But here's the tricky part. After the attackers are defeated and the king has won, he might try to pretend like he was defending both the Motte and the Bailey the whole time! He might say, "I fought bravely for both parts of my castle and saved all my people." This is called the Motte-and-Bailey argument.
The king is pretending like he was defending the Bailey, when in reality he was only defending the Motte. This happens in arguments too. Someone might make a big, strong argument (like the Motte) that's very hard to attack. But then, when someone starts to criticize it, they retreat to a weaker argument (like the Bailey) that's easier to defend. They bounce back between the two arguments, pretending like they're both the same thing. This is a trick that people use to win arguments, but it's not always honest.