When downloading a video is legal
This is general information, not legal advice, and the answer genuinely depends on where you live. The broad shape is consistent enough to be useful.
- Your own uploads and Creative Commons material are the safe cases
- Personal, private viewing is treated very differently from sharing
- Redistribution is where almost all real trouble starts
Content you made
If you uploaded it, downloading your own file back is uncontroversial. This is one of the most common legitimate uses — recovering a master you no longer have locally.
Creative Commons and public domain
Plenty of material is explicitly licensed for reuse, and archives like Archive.org exist for exactly this. Check the specific licence, since some require attribution and some forbid commercial use.
Personal copies of other people's work
Many jurisdictions treat a private copy for your own viewing differently from distribution, but platform terms of service may still prohibit it. Terms of service and copyright law are separate things, and breaching the former isn't automatically the latter.
The bright line
Re-uploading, selling, or publicly sharing someone else's video is where the serious risk lives, in essentially every country. If you wouldn't put your name on it publicly, don't.