What you are talking about is an API. So you'd build out the entire infrastructure on the first website, then on the second website, it would make some kind of calls to the first website to get back the information it needs, usually using some kind of credentials or access token.
This way, you can allow anyone in the world to communicate with your website, kind of like how Facebook, or Twitter does.
As far as accessing your database, you would need to tell your app somewhere the credentials to use, so technically, you do need to hardcode them somewhere as they can't just magically make up some credentials somehow to access a database.