Glad you asked - I was going to post something about it. Currently "kicking" is only available to Moderators, but it may get opened up to Editors or high-points people later on.
For Moderators, clicking the grey dot next to a username will kick them off chat for 10 minutes. It says "nominate" because in future it will likely take multiple nominations to kick a user (mods will always be able to straight kick them).
When a person is kicked, a message is posted in chat telling everyone else. It will have a pale yellow background.
There is still other stuff I need to finish, like the nomination thing. And possibly kicking people for longer if they are kicked multiple times.