Hero's Realm and Sword Trial
Multi-stage boss fights, each stage a single boss with mechanics your team has to learn and overcome before moving on to the next one.
The fights themselves
Both are multi-stage encounters, usually two stages, each stage its own boss with at least two core mechanics the team has to adapt to and overcome. Clear one boss, move straight on to the next — the whole thing is one continuous run, not separate, independently-queued fights.
Hero's Realm
A 10-player fight. Its weekly clear reward doesn't cost any energy, and there's no limit on how many times you can clear it in a week — but the reward itself can only be claimed once per week, no matter how many extra clears you rack up beyond that.
Sword Trial
A 5-player fight. Its rewards cost 60 energy per claim, but there's no weekly cap the way Hero's Realm has — clear and claim as often as you like, as long as you've got the energy to spend on it.
Different bosses every season
The specific bosses change from season to season, as part of the Boss Talents achievement below. Clearing these bosses and completing their individual challenges is how you work toward the Boss Talents achievement and its One Sword Holds Ten Thousand title.
Joining a run
Guilds using Guild Ledger Bot can post Hero's Realm and Sword Trial as RSVP events right in Discord — sign up for a role and build, and the event handles the rest.