microsoft sql express 2012 authentication connection issues - can't login
More information would help here and I've asked those questions in the comments. That said - a few thoughts:

Basically to access a SQL Server database you need a few things:


A login - this gets you into the server itself. This is not a database user, but a login at the instance level (In SSMS this is the top most security, not the security under a database.)
That login being mapped to a user - now there are exceptions but for the purposes here - you need that login to then be granted access to the database. This would be security found at the database level.
Permissions - you'll need permissions of some sort to do something in SQL.
SQL Server needs to have the protocol enabled that you are trying to connect with. Also if there is a firewall, the port (default is 1433 for TCP port) needs to be open.


When there are connection issues like you describe it is normally around something like not actually being on a domain but trying to use windows auth, a password mismatch, not having access to a default database selected for a login or an actual network issue like a firewall.

These are two older posts but they are fantastic resources for troubleshooting connectivity to SQL Server. This first link is from the SQL Server protocols team and describes the steps to troubleshooting connectivity issues. This second link is from the SQL Server connectivity team blog describing the various login failure messages and their states and how to troubleshoot each state.