The media library on LDS.org has a section of Social Media Shareable Videos.
This is a collection of short videos that have been published on Church social media pages.
This is a great library of videos you can share with others on social media. Below are a few of my favorites:
Strong Family, Strong Nation
Blessed with Great Joy
Challenges Strengthen Us
Find many more shareable videos like this in the LDS.org media library section of Social Media Shareable Videos.