Not sure if I understand you correctly, but...
The application that accesses the database needs a client library in order to communicate with the database. In your case this would be .Net framework since it contains
System.Data.SqlClient[
^] namespace which provides the db connectivity.
The database side on the other hand is built by installing SQL Server (Express in your case) on some computer and by configuring it properly so that the client has proper user information, privileges and network access to communicate with the DB.
In many situations the database server and the client application resides on different locations. This way multiple clients can access one, centralized database. Of course depending on the application the application and the database may also be on the same computer.
What comes to the version to use. Based on your connection string you've used Express edition which is free to use. Other versions require licensing except compact edition which can be embedded to your application. However, compact edition can't be used by multiple client.
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