Why does my router stop working when somebody goes on Youtube?

It is not your router that is at fault. I run three computers from one router at the same time and they all work fine. The problem is most likely that you (or your brother) is exceeding your download limit. Watching videos on YouTube counts as a download and you only get so much download per month whatever package size you have with your broadband provider. So you either need to upgrade your monthly download limit or stop using YouTube so much.