Is it possible to connect LOCALHOST DatabaseEgnine without providing the computer name like MyComputerName\LOCALHOST ?
(I use SQL Server 2012 - Standard Edition installed)
MyComputerName\LOCALHOST is named instance called LOCALHOST on your computer. To connect to it you must either specify the instance name (and have the SQL Server Browser service running) or specify the listening port explicitly (and have the instance configured to listen to static port, non-default since default is dynamic ports). So any of the below will work:
.\LOCALHOSTlocalhost\LOCALHOSTlocal\LOCALHOST127.0.0.1\LOCALHOSTmachinename\LOCALHOST.:<port>localhost:<port>127.0.0.1:<port>Is very unlikely you actually wanted to install a named instance called LOCALHOST. You probably didn't read the setup dialogs and misconfigured the instance name. My recommendation would be to install it again, choosing a default instance (ie. unnamed) or a more convenient and less confusing instance name.
.or(local)as identifier for local default SQL Server instance.