The simplest and quickest way to see if and what edition was installed is to run SERVICES.MSC and check for the installed service. To get to SERVICES.MSC, open
Control Panel ➮ Administrative Tools ➮ Services
then scroll down the list looking for SQL Server.
I have installed both Full and Express editions of SQL, side-by-side. The screenshot below shows what to look for. The Express edition is denoted by an instance name of SQLEXPRESS.
Notice at the right, I have SQL Express Disabled and MSSQLSERVER is Running. You can start SQL Express here if it's not running. You can also set it to start automatically on boot.
To launch SQL Server Management Studio, you'll want to look for it on your Start screen. It may look something like:
If it's not there, you can get the 2005 version directly from Microsoft here, or the 2008 version from here. You'll probably want the 2008 version.
Finally, you could also go for SQL Server 2012 Express here, which includes Management Studio Express, however, you'd probably have to uninstall whatever Express edition you have installed now.
I hope I haven't muddied that water for you. I'd be happy to clarify or answer any additional questions you have.