I am not sure of the software used on both the forums I help moderate; but I believe it's VBulletin.
On the first one, I have to deal with a lot of spammers. The admin (they're higher than moderators) are the only ones who can ban people. We seem to have a lot of spammers. As a result, your first post is moderated.
The second one, again, there's an issue with spammers, trolls and alter egos. The forum is far bigger (it's one of the biggest online) and all moderators are involved in their own area of the forum. For example, I help moderate the money and finance part and someone else moderates the health and relationships forum.
The second is also concerned with expanding the website and makes a lot of changes to the forum and the whole site have a say in what we do and don't do.
As a newbie moderator, I am on 3 months probation. First one done and two more to go. We have reviews to see how we're getting on, etc. I've made a few minor mistakes; but someone has kindly talked to me and explained what I am meant to do and what I shouldn't do.