In the first part of the "How to Backup Cisco IOS Configuration (GNS 3 Lab)" YouTube video, the speaker shows how to backup configurations using FileZilla Server and Client. This involves installing and configuring the software, establishing a connection, and using the "archive" command to create a backup. In the second part, the speaker demonstrates using the Cisco IOS command-line interface to create backups with the WR (Write Memory) command, which allows users to specify a time period and view saved files using the show archive command. The speaker provides an example of creating and viewing a backup file, and concludes by changing the hostname and creating a new backup.