One really statsy video from The Economist that I've featured previously on this blog explains the real life implications for Type I/Type II error in research (and, specifically, how it leads to errors in published research).
The other videos may not be as directly related to the teaching of statistical topics, but they do illustrate data. Topics range from American union membership trends to this video about world population growth. As you may have inferred from the source, many of these videos focus on national and global economic information, but all of the videos do present data that you can integrate into your classes.
Some are more applicable to teaching statistics: This video describes why we have so much data and keep on generating more data. Others are particularly applicable to social/interpersonal psychology, like this illustration of how "like likes like" in terms of education level (and how this may contribute to income inequality).
|Like likes like|
These videos demonstrate to your students a) how data can be used to make a logical argument, b) how illustrated data can help to visualize a compelling story, and c) that statistics are used by people who do not work in explicitly stats-y careers.