Cannot connect to local SQL Server with Management Studio

Same as matt said. The "SQL Server(SQLEXPRESS)" was stopped. Enabled it by opening Control Panel > Administrative Tools > Services, right-clicking on the "SQL Server(SQLEXPRESS)" service and selecting "Start" from the available options. Could connect fine after that.

I was having this problem on a Windows 7 (64 bit) after a power outage. The SQLEXPRESS service was not started even though is status was set to 'Automatic' and the mahine had been rebooted several times. Had to start the service manually.

Check the sql log in the LOG directory of your instance - see if anything is going on there. You'll need to stop the service to open the log - or restart and you can read the old one - named with .1 on the end.

With the error you're getting, you need to enable TCP/IP or Named pipes for named connections. Shared memory connection should work, but you seem to not be using that. Are you trying to connect through SSMS?

In my log I see entries like this...

Server local connection provider is ready to accept connection on [\\.\pipe\mssql$sqlexpress\sql\query ]

As the comments said, .\SQLEXPRESS should work. Also worstationName\SQLEXPRESS will work.