New to meditation. I'm confused?

I've just started meditating and reading about it and I am slightly confused about your thoughts during meditation. Many people discuss letting your feelings and emotions come over you and then letting them go; however, I was under the impression that we were not supposed to be NOT thinking and the goal was to literally clear your mind of all thought and just hone in on your breathing. Am I wrong? Or are these two different types of meditation entirely.

When I have attempted mediating I will lay down, focus on my breathing, and whenever a thought would come up I would basically repress it and just try to think nothing. Only In......Out. How would I even go about riding my emotions if I am supposed to be thoughtless? lol