When you're a community manager, you generally problem solve and do customer service and assume the best of people.
For example, Google uses forums to do customer service, which is frustrating, but a friend of mine went over there, said, "Hey, I'm frustrated! I need my google adwords account reactivated, for some reason it's blocked, what's the deal, I really want to give you money, etc" and within 24 hours they got him back on track!
This is an example of someone who's not trolling. He's now an advocate of their forums, because they actually responded. And a lot of times, that's all community members really want you to do, respond, even if they are frustrated. When you ignore their frustration, it just makes them angrier.
How to tell when someone is a troll
When someone is trolling, you can tell, because there will be a certain illogic to their actions, and they might try to point you towards various websites with porn on them, or they might attempt to direct you to a Rick Astley video.
Trolls are fond of trolling the instant message service of the Church of Mormon with queries such as, "Magnets, how do they work?" This level of trolling is fairly harmless and you don't need to worry about it.
Trolling does occur with such frequency, however, that there's even a set of memes devoted to trolls, such as "Art of Trolling" on Memebase. I have put the classic "trollface" into the picture below.
When Trolls Attack
If they send more than 3 insulting messages, if they start threatening you, your company, or just basically harrassing you with various kinds of written abuse or horrible pictures, time to get medieval on their handle!
Don't try to reason with the troll. They are taking advantage of the fact that they will never meet you, and so can say anything they want to you. The advent of the internet has really not helped with common courtesy, unfortunately. So you want to get your community members to sign a little statement saying, "I will not post blatantly self-promotional things, I will not troll or flame or spam people or be obnoxious in any way (As defined by the community manager) or I will be blocked and banned."
What you can do for this sort of negativity
Once you have identified a troll (which will be a very very small percentage of your users, you can instruct other community members "not to feed the troll". Then, like the three fates, you can measure out their time with your community, and then you can cut and block them from communicating. You can block people easily in Twitter, and you can also ban their IP address from accessing your website. This might inconvenience a few other community members, so check with your IT person to see if there are a lot of people on that IP address accessing the site.
I hope this was helpful, and if you have any other stories of what you did to overcome trolls, or deal with negativity, I'd love to hear them!