EDIT
It has been pointed out to me that a link in my answer does no longer work. Which is not that surprising, since this is a 3 year old answer.
Because It seems like people are still stumbling on this answer, I would like to redirect people to this page:
https://gorails.com/deploy/ubuntu/16.04
It should have all the information you need to deploy a rails application.
End EDIT
I recommend that you check out
http://rubyonrails.org/deploy
to start with.
And also
http://www.cloudfoundry.com/
and
https://www.engineyard.com/
are both great hosts for any rails application, and they also have some pretty easy to follow step-by-step tutorials on how to do so.
As Leito mentioned Heroku in his answer, I would like to point you to
http://www.codeschool.com/code_tv/heroku
If you choose Heroku as your host, you could check out that video as it shows you how to deploy a simple rails application.
