Permission denied on my host

My <stack> app on <host> fails with a permission error:
<paste the error>

Explain for this host:
- Which user my app actually runs as, and which user owns the files.
- The permissions this path needs, and the smallest change that works.
- Why "make it 777" is the wrong fix, and what to do instead.
- Whether a folder the app writes to should be inside the web root at all.

Then give me the exact commands, or control panel steps, to fix it.