What is the best way to name a kind of work when you do your job from home? (For example, I want to write an essay about it.)
You want to use the second version of the phrase.
I work remotely.
Here, remotely is an adverb and the sentence means that you are working away from the office (or away from wherever the main work site actually is).
I work on the remote.
Here, remote is a noun and the sentence means that you are working on something like a remote control—most likely repairing it or figuring out how to program it.