To change the order of fields, grab a drag-and-drop handle and drag the field to a new location in the list (grab a handle by clicking and holding the mouse while hovering over a handle icon). Remember that your changes will not be saved until you click the Save button at the bottom of the page.
The order you define will be used both on input forms (when creating or editing a post), and on content display (teasers, content page, RSS items...)
You can also change the order of non-CCK 'fields' like Title or File attachments. Depending on the 'field', this will affect input forms and/or content display (some of those 'fields' are not displayed in both contexts).
If your content type has groups (requires the Fieldgroup module), you can move a field inside a group by dragging it below the row of the group, and then slightly to the right before dropping it. Note that groups can also be reordered, but can currently not be nested inside other groups.
When adding a field or a group, you can drag them directly to the intended spot in the list of fields and groups that are already present in your content type, before clicking Save: