A unit die when it either get as much damage as it has HP (or its HP get reduced to 0 via -1/-1 runes or effect) or that an effect instruct you that it dies (like sacrifice or illusion being targeted).
Yes sacrifice count has dieing.
You can sacrifice unit you control (which is different of owning).
"Dies" and "sacrifice" are both in the glossary at the back of the rulebook. "Dies" is when something moves from "in play" to your discard pile. You can only sacrifice stuff you control (it would be kinda silly if you could sacrifice something belonging to an opponent).
There's a theory: if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable. There's another theory: this has already happened.
I think you can also only sacrifice units if a card or effect requires you to do so.
Or could you sacrifice any own unit in play at any time?