The word "rut" means a hole that's long and thin, like the hole that a wheel leaves when it rolls through mud.
"Rut" is most often used in the phrase "stuck in a rut". Being "stuck in a rut" means that you keep doing the same boring thing each day. It sounds depressing. Use this expression if you feel like someone's life is boring.
You're stuck in a rut. You need to get out more and try something new.
You can also use "stuck in a rut" to describe your relationship with someone like your spouse, boyfriend, girlfriend, etc. In this situation, it means that you don't do interesting things together any more.