There are many ways to write the connection string, but below is one example.  In this example 'ServerA' is the principal, 'ServerB' is the mirror, AdventureWorks is the database when using Windows Authentication to connect to the SQL Servers:

"Data Source=ServerA;Failover Partner=ServerB;Initial Catalog=AdventureWorks;Integrated Security=True;"

Below are some more helpful links that might answer most of your questions\concerns.

http://www.connectionstrings.com/?carrier=sqlserver2005

http://www.microsoft.com/technet/prodtechnol/sql/2005/dbmirror.mspx

http://msdn2.microsoft.com/en-us/library/ex6y04yf(vs.80).aspx (What's new in ADO.net 2.0)

http://msdn2.microsoft.com/en-us/library/5h52hef8(VS.80).aspx  (Connection String Info)

http://msdn2.microsoft.com/en-us/library/system.data.sqlclient.sqlconnection.connectionstring(VS.80).aspx (SqlConnection.ConnectionString Property )