Greetings.
My name is Ryan, and I've been actively moderating Stack Overflow since January 2015. I actively participate on meta, and I'm interested in keeping Stack Overflow awesome. As my many MSO answers show, I've become very familiar with how Stack Overflow works, and I eagerly try to help others get the most out of the site.
My moderation style focuses around communication. I believe that, more often than not, users are trying to do the right thing, but that they often can't express what they're trying to accomplish. Proper edits and timely comments can get posts back on track and resolve conflicts before they get out of hand. Likewise, as an RO, I would first try to diffuse situations by making sure that everyone is on the same page. I am not shy in editing posts to make them more understandable, especially to fix grammar.
First, I try to understand exactly what the user is actually saying. Then, I investigate the claim and try to figure out what happened. My next actions depend on a few things:
I was a bit late to this particular argument, but I do believe that my summary that the two disagreeing sides actually were arguing for the same thing helped to bring the argument to a close. More generally, I also comment/edit meta posts that have unpalatable wording likely to attract downvotes.
Often, questions askers on meta don't fully understand how Stack Overflow works, so I try to get to the heart of the misunderstanding and keep discussion on-topic and helpful. For example, I correctly interpreted and brought this meta discussion back on-topic.
A room owner is a representative for the chat room. They are the people that others will look to first for determining the culture of the room. Additionally, as the SOCVR home page describes, ROs also are responsible for keeping everything in order. This includes housekeeping stuff like cleaning up processed cv-pls and more serious duties like chat room moderation.