The connection string you need will depend on several factors, such as the edition of SQL Server (LocalDB/Express/Standard), whether or not it's a named instance, and the type of authentication (SQL vs Integrated) you have in place.
The answer from @zippit is a good example of a connection string for LocalDB using integrated security.
The same string to a Sql Express server would look like this:
"Data Source=serverNameOrIpAddress\sqlepxress;AttachDbFilename=|DataDirectory|\mydbname.mdf;Initial Catalog=mydbname;Integrated Security=True"
..and to a standard edition of Sql Server would look like this:
"Data Source=serverNameOrIpAddress\sqlepxress;AttachDbFilename=|DataDirectory|\mydbname.mdf;Initial Catalog=mydbname;Integrated Security=True"
..and to a named instance on a standard edition of Sql Server would look like this:
"Data Source=serverNameOrIpAddress\instanceName;AttachDbFilename=|DataDirectory|\mydbname.mdf;Initial Catalog=mydbname;Integrated Security=True"
All those assume integrated authentication. If you have sql authentication set up, you would substitute "Integrated Security=True" with "User Id=username; Password=pword;"
Also, if the sql server is on the same machine, you can use this for the Data Source parameter for Sql Express
.\sqlexpress
..and this for standard Sql Server
(local)
Here's a site I've found useful: Connection Strings