Hi Hytaler,
Hardcore landed with Update 6, together with Spectator mode, and the two belong together. Here's what each mode does and which one I'd pick for which kind of group.
Where you find it
Hardcore is set when you create a world: hit Customize, then the Hardcore dropdown. Life counts can be adjusted (anything but zero). It's a world setting, so decide before you start, not halfway through a run.
The modes
Off. Normal Hytale.
Permadeath. Every player has one life. Die and you're out. The classic, and brutal with friends who like to explore ahead.
Three Strikes. Three lives each, then you're out. My pick for most groups. You still feel every death, but one bad jump into lava doesn't end someone's evening ten minutes in...