Where are all the references to the URL stored?
Craft makes it easy to handle multiple environments, but you need to make it aware about the differences. Usually it's smart enough to enough to know the base folder path that you have it hosted in.

It's also smart enough to usually which url's you want. Sometimes it doesn't properly remove index.php from the URL, and that you can manually override with the setting omitScriptNameInUrls.

Since your public folder is named public_html, it's not clear from your question if you are running with the standard folder structure (index.php located in public folder, $craftPath inside index.php pointing to craft/ folder one level above public folder).

In general, you should have a look at the documentation regarding multi-environment configuration.

The one other area where it's usually necessary to set environment specific settings is Asset sources: Check out the chapter on that in the docs as well.