Room-wide settings live in the chat options panel when you're viewing your own chat room - only you see that section, and moderators cannot change your room settings. Actions on one person (e.g. granting mod, kick/ban, etc) are accessed by clicking their avatar or username.
Sets how long someone has to wait between messages. Off by default, then 5s, 10s, 30s, 1m, 2m, 5m, 10m, 20m or 30m.
Useful when a room is very busy, or as a first line of defence against spam.
Allow new users to chat is on by default. Turning it off restricts talking to accounts whose email was confirmed at least two weeks ago.
Allow anonymous users in chat controls whether people who are not signed in can join and talk. If new users are not allowed to chat, this is implicitly disabled and non-signed-in users will not be able to speak.
Use moderated chat is the strictest option. With it on, only moderators and people you have explicitly given voice permission can speak. Everyone else can only read.
Room link policy has three choices:
Every message carries a delete button and a report button. Moderators can delete any message in the room. Whether an ordinary viewer can delete their own is the Allow message deletion in room setting - moderators can delete messages either way, so turning it off does not limit mods.
Clearing the whole history is a separate command,
/clear. It clears the room for everyone if you're a mod or streamer, or just for yourself if not.
Kick removes someone from the room. They can rejoin, so it works as a warning.
Ban removes them, stops them coming back, and deletes their messages from the room.
Unban lifts a ban.
In a room using moderated chat, only people with "voice" can speak. Admins, Streamers, and Moderators all get voice by default. Grant voice and Revoke voice control that, one person at a time.
Streamers can Grant mod status and Revoke mod status from the same card. Moderators cannot appoint other moderators.
A moderator can kick, ban, delete messages, set the topic and clear history. They cannot change your room settings.
You do not need to be a moderator to stop seeing someone's messages: if someone is annoying you block them.