The development environment is obviously for developing the web app locally.
The production is the operational public website with everything as the final version.
what is the testing environment and what is the difference between that and staging environment?
and if its what I think it is, why not just deploy to an alternate website address and use that?
The testing environment is an environment in which Rails will perform the tests that you write. This is a slimmer environment in which certain things are disabled such as sending emails, and loading the asset pipeline. The database in the test environment is also generally not persisted between test runs.
For more information on how and why to test your rails applications check out the Rails Guides. If you are not currently testing your applications, you should really start as it can save days of hunting for broken code, and many many headaches down the road.
Staging environments are setup to mimic most of the functionality of a production environment. The staging 'stack' should be as similar to the production environment as possible; e.g. same hardware, same OS, same installed packages, etc.
Things I generally limit when setting up a staging environment are email sending, I will use a mail interceptor to send all email to my address. Think of the staging environment as a sandbox in which to test changes to your production environment, without actually testing in production.
For most small apps, a staging environment is not always necessary. However, for a large, enterprise level application it is a must as you generally cannot afford any downtime for your production system. With the staging environment you can catch those changes before they are pushed to production.
Hope this helps.