Missions are similar to Challenges in that they use your cumulative effort to complete but are different in that they generally apply to multiple stages or events. For example, the Tour Italy Challenge will have you complete 2000m of cumulative cycling while the Watch The Femmes Mission had you complete 8 different stages (group ride events aligned with the Tour de France for women). Zwift has also been more proactive recently in creating more missions while new challenges have not been added in several years.
Example of Mission Blocks on Home Screen
New missions are announced in Zwift via emails, social media, companion app and as a new block on the home screen. You generally have to register for the mission prior to starting any of it's stages. Registration is easy by simply clicking on the mission block on the home screen and clicking 'register'. While you are in the process to complete a mission, you percentage progress to completion will be shown when you click the mission block on the home screen. The companion app will also show your percentage progress as well. Mission are also time constrained and must be completed by the final date in order to be considered complete.
Completed missions award you a Mission Badge and often unlock virtual gear (jerseys, socks, headgear, bikes, etc) periodically throughout the progress of the mission. For example, at 25% of mission completion, you might receive a virtual pair of socks. Then at 50%, you might unlock a virtual headband and finally at completion, unlock a virtual jersey and be awarded the Mission Badge. Mission Badges are shown on your Achievements screen at the very bottom of the list. Missions are listed by year completed.
Example of completed Mission Badges on Achievements screen