What make of router is it? If you trust your computer man, you don't really need to worry about changing the password.
I leave my router on, but it is perfectly ok to turn it off. Many people advise this - not only because it saves electricity, but it gives a little more security - nobody is going to rely on your wifi being on 24/7.
You really should take the time to understand your router settings though. It is nothing to be frightened of, and will help you sort out future problems.