https://www.wired.com/story/this-prosthetic-limb-actually-attaches-to-the-wearers-nerves/

Most prosthetic arms use the person’s other body parts, like the shoulders or elbows, to power them, which limits dexterity. But the one the team demonstrated in the study was attached directly to a user’s own nervous system, allowing him to individually move each prosthetic finger at will.

The article is mostly about the newer tech, but at a couple points it goes into the basics of the older tech, if that's what you're curious about.

Depends on how much they can do. If they can attach the nerves or muscle to something, you can use them similarly to how your real arm works (it's not perfect and you have to learn how to do everything with it again but it's still pretty damn cool). If they can't do that, like there's just too much damage, then you'll probably get something you have to manipulate with your other hand or just looks nice but has very limited functionality.

The old school prosthetics tended to be strapped on in a way that let you use other muscles to operate the grasping part. The typical ones you'd run across were hooks. They had two sides, and you'd essentially flex your upper arm to spread or close them (over simplifying, but that's the gist).

And that's in terms of stuff that did move. Some were purely aesthetic, others (like the classic pirate hooks) served to allow a range of tasks, but didn't move. There's some examples of really old prostheses that were remarkably dexterous, but they tended to be one-off items.

In terms of the full on mechanical hands, those are relatively new. They've been developed for decades, with a handful of methods used to try and bridge the gap needed what the person wants to do and the movements the device makes. Anything from picking up brain signals to electrodes that detect muscle twitches at the site.