granting access for remote account to local database instance
You are going to want to create an AD service account on your SQL server. Then also make sure that AD service account is on application server, and is you local admin group on that box (I'm guessing on this part).


Create the AD Service account
Add the newly created AD account to your SQL Server, but wait a bit for the AD account to propagate. Give it a minute or five.
Create a new SQL login, keep Windows Auth selected, and click the Search... button 
When the Select User or Group window appears, click the Locations... button If you are wanting to add a group, then you will want to click the Objects Type button, and select the Groups checkbox before you search

On the Locations window select Entire Directory, and click OK

Enter in the user name you just created, and click the Check Names button. If it finds the user you will know, and click the OK button
Now, assign it to a server role, or assign it a database with the permissions needed for your application.


Now try to connect your application server to your SQL Server.

Now, you can cheat. However, I wouldn't recommend this, but if you don't really care who is making the call from your application server. Run this script below. Change the domain to your domain, and replace SERVERNAME with the name of your server. Once this created on the SQL server give it perms to the server or database in question.

 CREATE LOGIN [DOMAIN\SERVERNAME$] FROM WINDOWS WITH DEFAULT_DATABASE=[master], DEFAULT_LANGUAGE=[us_english]