A gentleman's agreement is kind of like when you make a promise with someone just by using your words, without any official papers or handshakes. It's like when you tell your friend that you both will share your toys and be nice to each other when playing together. You didn't sign anything, but you both agreed to it, and you feel like you should keep your word because it's the right thing to do.
In the grown-up world, sometimes people make a gentleman's agreement when doing business. For example, two businesses owners might agree to not sell their products in each other's areas. They don't write it down or sign anything, but they both know what they promised each other. It's like a secret promise that they made to each other.
The reason why people make gentleman's agreements is that it's a trustworthy and respectful way to do things. People feel like they can rely on each other, and it helps build good relationships. However, these kinds of promises might not have any legal power, so you always need to be careful and make sure you know what you're agreeing to.