Handbook -- Chat Mod
Rule 1: follow the rules. You have to follow the rules better than anyone else in chat. It's practically impossible to enforce a rule you just broke, so don't back yourself into that corner.
Have you looked over the Moderator's Promise? You may not think it's overly important, but a) it was written as an ethical guide by the founder, b) that founder spent a lot time in a downright crazy chat room as a mod, and c) other experienced mods have looked it over and given it a passing grade. It's a good place to begin understanding the kind of character it is hoped you will display.
Chat is meant to be a fun place, for the mods as well as the chatters. Don't be afraid to joke and have fun, just don't do it at someone else's expense.
If ever a situation arises where you don't think another mod should be saying something or doing something (and I am not talking fireable offenses here, just small stuff. If there is a major problem, take it straight to the chat admin) never, never, never call them out in main. Open a pm and talk to them: be courteous, explain why you think it's a problem, ask that they stop. If you "correct" them in main, you humiliate them, undermine their authority with the other chatters, and may just cause more problems than you solve. If there is a consistent problem, record it, send it to the chat admin.
Don't be afraid of confrontations. You were chosen for this position, which means we will trust your judgment. Deal with people disrupting chat, and the rest of the mods will back you up.
Keeping notes on people in chat isn't bring stalkerish: is actually one of the more useful habits you can form as a mod. Screenshots are also your friends. As a chat mod, you blend into chat in a way admins can't. People accept you as normal, so they act normal around you. Your notes on character can help during nominations to staff, can give another member of staff an idea of who they need to keep an eye on, or remind you which people you need to keep an eye on. Added bonus, it makes keeping track of fifty people and conversation topics easier.
Pay attention to new users. Best case scenario, you're going to be helpful to them in getting used to the Library atmosphere. Worst comes to worst, something about the first encounter will often tell you more about a person's character than you can find any other way. The worst trolls identify themselves to an attentive mod before they strike. First impressions matter.
Kick away. Seriously. A kick is a warning (which should probably be preceded by a verbal warning in most cases). If you warn verbally too many times before kicking, you weaken the impact. Always try to be fair, but don't hesitate to kick a problem.