At the bottom of any YouTube page (under Countries and Languages) you will see a Safety Mode option. Safety Mode is an opt-in setting that can either be turned on or off at will -- or it can be locked on by default in your choice of browser (you'll have to make the opt-in choice in each browser you use). YouTube created a video (what else) to explain the process.
So what kind of videos does Safety Mode filter out? Think about things like news coverage of violent events. This content is within YouTube's Community Guidelines, it might not be something that you (or your children) want to see.
If you don't have Safety Mode right now, don't fret -- YouTube says it is rolling it out to users throughout the day.
We like that Safety Mode is opt-in, rather than something you have to seek to disable. It's also nice that YouTube has decided to add more controls to the overall experience.
Now if only they could create a safety mode for stupid comments...