The relationship and difference between a Google+ Page and a Google+ Community eluded me at first. After having created a few of each for the various user groups I’m involved with, I think I have a handle on how to set them up.
You start with a Page for your group. Make sure you set up Managers for the Page. This can be done by switching to the Page, then clicking the tiny Settings link in the lower right, and then on the Managers tab.
From that Page you create a Community. See, a Page is like a person (kind of like a corporation is a person), so it can own Communities. Those people who are Managers of your Page can also manage the group by switching to the Page first. The Page will be the default Owner of the Community, but you can add other Owners. If you’re viewing the Community, click on the Members link. You can then use the little triangle next to member’s names to promote them to Moderators and from there to Owners.
Hopefully this will get you off on the right foot with your Pages and Communities.
[Edit]
By request, here are the Google+ Pages and Communities I’ve setup:
