When granted moderator status in a community you are authorized to do the following,
A settings button should appear right above the listed members on the right hand side. From here you will have access to the following settings.
Add another community member as a moderator. Under the moderators tab.
Invite new people to join your community. Under the Invite tab.
When clicking the three dots on the top right of any post in your community. You will now have two additional options.
"Pin" Which pins the post to the top of the community feed.
"Remove from community" which removes the post.
Click on members, then the three dot menu on the top right of any person. You will then have the option to remove this person from the community.