What is a trusted connection in terms of SQL Server 2005 (Trusted vs Windows Auth)?
A trusted connection is the same thing as using Windows Authentication in SQL Server 2005. Authentication is done by the domain, and authorization is handled by SQL Server.
SQL Server can also use its own logins, such as the sa user. These are both authenticated and authorized by SQL Server. They are only viable if SQL Server is run in Mixed Authentication Mode.
MSDN has a good article about choosing the right Authentication Mode. It's a good primer.