The default rails server when you run it on your local computer is WEBrick, which is great for developing, but awful once a bunch of people start using it at once. Instead, let's add a new, more production-ready server in on Heroku, and add a file that tells Heroku to use it.

Gemfile

source 'https://rubygems.org'

group :production do
  gem "thin"
end

Procfile

web: bundle exec rails server thin -p $PORT -e $RACK_ENV

I'm recommending a production server called 'thin' because it's well-tested and very reliable. Other servers exist, and if you want to research and experiment, you're welcome to.

To get this working, you'll have to add a new file in your root directory, and this will live right next to your Gemfile & Rakefile. You must name it 'Procfile', with no quotes and no extension – that's the file Heroku expects to see. Store that in your root directory, and make sure it's committed when you push to Heroku.