What is a paradox? Something that is unsolvable and just exists to infuriate us? Maybe, but it certainly is not to philosophers. A paradox is a problem that clashes the definite ideas you obtain from life. The definite idea can be a mathematical formula or a simple truth like the distance from A to B is finite. The things we know for sure like the distance example can be blurred by a paradox.
A philosopher in Ancient Greek takes a walk after a tiring day, only to take some fresh air. At that moment he comes up with a rather interesting theory. He says that the distance I walk takes a finite value of time, the half of the distance again costs us a calculable time. If the distance is to be halved after and after, we would have infinite values. And that brings us to the conclusion which the time needed to go from A to B is infinity. Hence motion is an illusion.
But how can that be true? I can walk from my house to a grocery shop in 10 minutes. And that is my fellow readers a paradox in action. The reality and the stated phenomena are up against each other. Some might make a reasonable explanation about why this paradox is falsidical and you would be probably right. Because many of the paradoxes are refuted like this particular one.
The philosopher made a mistake by accepting that infinity numbered finite values cannot be equal to a finite value. That is the reason why the conclusion is both unrealistic and somewhat logical at first sight.
On the other hand, some paradoxes seem impossible but true such as Monty Hall Paradox. You are on a TV show and competing for a prize. There are three envelopes in front of you, you choose one then the host opens one of them, usually the false envelope. Then you are given a chance to switch or not. Long story short you double your chances by switching. This is a type of paradox which is true.
Another type of paradox is popular The Grandfather Paradox. In this one, you travel back in time and kill your grandfather. Because you kill your grandfather your father is never born so are you. Well if you do not exist how can you kill your grandfather? No matter which verdict you end up with, these paradoxes cannot be true and cannot be false.
As given examples show that paradox is sometimes unsolvable and sometimes needs to be proven wrong. In every scenario, they all start with a curious mind and the perfect conditions for it.