Do you know what video game consoles are? They are machines that you can use to play video games on your TV. Examples are Playstation, Xbox and Nintendo.
Okay, so the console war is when these companies (like Playstation, Xbox and Nintendo) are competing against each other to sell more of their video game consoles. They do this by advertising their consoles and trying to convince people to buy them instead of their competitors'.
Just like when you and your friends want to convince your teacher that your favorite color is the best, these companies want to convince people that their console is the best. They might try to do this by showing people how great their games are or how powerful their machine is.
The ultimate goal is for the company to sell more consoles than their competitors so they can make more money and be the "winner" of the console war. However, it's important to remember that no matter which console you choose, the most important thing is that you have fun playing the games!