What is my database name?
Ok, there are a few things to explain first. 

(1) The Connect to Server prompt within the Login properties is only asking for authentication. You cannot specify a database name within the Server Name field of that window in order to connect to it. For the server name you specify either server or server\instance name. You would connect to a specific database by setting this via the Connection Properties tab:



(2) Folks that reference using the &lt;browser for more...&gt; option within the login portion should understand that in a default installation of SQL Server 2012 the Browser service is disabled. So even on a local installation you will not be able to see your server via this method.

Ok, now if you need to find out the "name" of the SQL Server instance you want to connect to you can follow any of the methods below (this is not an exhaustive list, just the most common ones to use):

SQL Server Configuration Manager

Via the Start Menu you would find SQL Server Configuration Manager and open it to look at the Services to see the list of SQL Server components installed on the server. You can also access this, if on the server, via the Computer Management console (you may actually see it show up twice if more than one version of SQL Server is installed, as there is on my  machine):




PowerShell

A simple one liner can provide the same information you will see in SSCM or the Services Management Console (services.mcs) if you are local to the server:



If you happen to be remote you can try to use this command but it will not work if proper access is not granted: Get-Service *SQL* -Computer Server1

To connect to the instances you find with SSMS if you see the services listed as SQL Server (MSSSQLSERVER) this is a default instance and if you are on the physical server you can use the server name, simply put a period (.), or localhost.

If you find a named instance it will be noted as SQL Server (instance name), as on my machine I have a named instance of SQL14 so I would put localhost\SQL14 since I am on my machine. If I had this on a server and was connecting remotely it would be the actual server name instead of localhost.