Ruby on Rails Hosts
For testing out your new application, Heroku works really well. It's free tier is good enough to run a Rails app which does not get too many users. The only drawback for the free tier is that if the site is not being hit for a while it take a 1-2 seconds for it to load. Its not a problem if visitors come to your site regularly.
Amazon provides a free VPS instance for year. Its a great place to run your application if you are an established company. But it might be a little expensive for hobbists or very early stage startups.
S3 is a great way to store assets like images, videos, any other static files. Its super economical, and easy to incorporate in to Rails.
Low end VPS
You can get a great bargain for a VPS at Low end box. While not suitable if you have thousands of visitors coming to your website everyday. A good way to get started and run computation tasks, scraping scripts or a CI.
Openshift is the Redhat cloud. Somewhat like heroku but was quite buggy when I tried using it.
Generally this is a big no no to run a Rails project, because the performance isn't too great. However, if you just want to run a personal blog it should be fine.