How Do I create a new user in SQL Server?

I've got a service user that needs to connect to SQL Server.  When I connect as myself, it works, but when I connect as this service account it doesn't work.  It looks like he doesn't have enough permissions.

I've created the login for the service.  It is set up in Active Directory, so I used Windows Authentication.  Server Roles I left as Public.  I clicked on the three databases that it would need to be connected to in User Mapping, and checked db_datareader and db_datawriter.  Is that enough to get the login access to read and write those databases?  

I see the user showing up in the users for each of the databases I selected.

Thanks,

Drew