How do @everyone and @here work? What's the difference between them?

@here notifies all non-idle users in the channel currently online.

@everyone notifies all users in the channel, even if they are offline.

If you can't mention @everyone or @here in a channel, then the server owner likely hasn't given you permission to do so. Server owners can enable or disable the option for specific user roles in their server settings, and in their channel settings.

To set permissions on a server, click the small arrow next to the server name.

Open Server Settings -> Roles, then click a role to customize that role's permissions.

To set permissions on a channel/channel group, click the small arrow next to the channel name.

It's worth noting that users can suppress @everyone and @here notifications on a per-server/channel/channel-group basis in their notification settings. See How do I stop @everyone mentions from select servers? for more information.