What does Mailman's "mod" (moderation) checkbox do?
Mailman's "Membership Management" screen shows the subscribers to your list and the various options set for each subscriber. One of these options is the "mod" (moderation) checkbox.
If a subscriber has the "mod" box checked, it simply means that the address will not be able to send messages to the list without the separate approval of the list owner or moderator.
For announcement only lists, all member addresses should have the "mod" box checked, except for the email address of the list owner (or other people who should be allowed to post announcements). If you initially create your list as an "announcement only" list, this will happen automatically: the owner will be added as a subscriber who does not have the "mod" checkbox turned on, and everyone else who joins will have the "mod" checkbox set.
This system is very flexible. If you want to allow a few additional people to post to an announcement only list, just uncheck the "mod" checkbox for the extra people.
Why do messages from the list owner or moderator get rejected?
If you're a list owner or a moderator, and your messages to the list get rejected with a message saying you're not allowed to post to the list, you've probably subscribed yourself as a "moderated" member of the list somehow.
To fix this, go to the Mailman membership management page for your address, then uncheck the "mod" checkbox and save the changes.