Setting up SQL Server to handle queries from external domains I am not a SQL Server admin. I have tried to google this but can't seem to use the right words to get the right info and they all talk about remote connections to a SQL Server in terms of admin and remote access. 

I have a database on one domain, that needs to accept connections and queries etc... from web code (page) I am hosting on another domain. I am using Windows 2012 servers on AWS in the case of both domains, and I have port 1433 opened on the domain server with the database, but open only to the ip address of this other domain. The SQL Server is using the default windows based auth.

Can someone tell me where to find a guide to set up the SQL Server to accept these external requests with a proper user name and password ?
It depends on what you mean by a proper user name and password. 

Windows Authentication

You will need to setup the Domain trust relationship between each domain. Your domain hosting the database server will need to trust the domain holding the web application. This should allow the domain accounts access the web application can connect to the database server, without having to use a separate account.

SQL Login

If you do not want to deal with the domain relationship you can simple configure SQL Server in mixed mode. You can find those instructions here. You would just create a SQL Login and grant the required access to the database(s) for the web application.