The authentication modes are:
Windows Authentication mode = enables Windows Authentication and disables SQL Server Authentication
Mixed mode = meaning that the server will accept both Windows logins and/or SQL ones
Windows Authentication is always available and cannot be disabled.
SQL Authentication requires that the user enter a username and password (or this information can be pulled from a file) or in some other way passed. This is good when multiple users need to connect to the SQL Server using the same username and password. For example a web application will usually connect via SQL Authentication.
Windows Authentication passes the security token from the Windows OS to the SQL Server so that the user is authenticated based on thier domain credentials.