Moments was a Twitch feature where channels could create a clip when something special happened. This would allow viewers to claim a Moments and upgrade their badge. The badge is only available in the channel where the moment happened.

Moments never made it out of beta and the help article now has the following note:

"The Moments beta will be rolled back on October 3rd, but don't worry you'll keep all badges earned during the beta."

https://help.twitch.tv/s/article/moments