SQL Server 2008 R2 can't connect to local database in Management Studio (Cannot connect to local instance)

Make sure that you are connecting via correct instance name. If your instance is SQLEXPRESS, then you need to use .\SQLEXPRESS or machie-host-name\SQLEXPRESS  as your server name.

And if you have a named instance then put that name as server name.

If this does not solve your problem then you need to check in services if your service is running or not.

Go to Run

 Type services.msc

Go to SQL Server (MSSQLSERVER) service.

Right click service and START

This should resolve your issue.

Write me if you are still facing same issue.