Yes, typically when testing web applications, you have a second instance of the application running on a different machine than the production server, configured to run against a different database server as well. The exact details of how to do this well depend on how your application is written and its requirements.

If all you want to do is functional testing of the application by yourself, then it is OK to deploy a copy of the software on your desktop along with a test database. I do this for the application I work on, and when I need to test against MSSQL, the Express edition is good enough.

If you need to do load testing or integration testing or other more complicated forms of testing, you usually try to duplicated the expected production environment as well as you can. This may not be possible, so you may have to make decisions about what's most relevant and simply accept those tradeoffs.

EDIT: Don't worry about CI or anything else that fancy and complicated for now. You may not even need it; that depends on your team and how often your application changes. Just for now take the simplest steps you can until you get this basic stuff figured out first.