Go to the Group Membership tab on the Edit User page to see the list of current assignments of the user.
Here you can move a user to a different organization. Just select it in the Organization menu.
You may add a given user to other groups by selecting corresponding checkboxes or remove this user from groups by deselecting them.
You may also create new groups in the same window:
Once you click the Add button, the new group will be added and you will see a corresponding notification.
To keep the changes you've made, click the Save button. The new user information will be saved, and you'll see a corresponding notification.