There are two ways to change a member's membergroup: One way is via their profile, while the other is via the edit membergroups page.
To edit the membergroup via the member's profile, go to members profile > Modify Profile > Account Settings. If you do not see this make sure that you are logged in as an administrator. Near the middle of the page there is a drop down list with the member's current membergroup selected. Change the box to display the desired membergroup and select "change profile" to save. You can also assign additional membergroups by selecting the "show additional groups" link below the selection box.
To edit the membergroup via the Administration center, go to the membergroup section (Administration Center > Members > Membergroups > Edit Membergroups) and select the name of the membergroup you want to place the member in. Here you can add members you want in the membergroup by typing their username in the "List of Members to Add" field and then selecting the "Add Members" button.
Notes:
- You cannot assign a person as a local moderator (moderator of a specific board) through either of these means. See How do I assign members as local moderators for more information.
- This process only applies to membergroups. You cannot assign members to post count based groups.