How to control the order in which the markup for fields appears in the outputted html source for content?
You can change tho order these fields are displayed from Manage Display (admin/structure/types/manage/CONTENT TYPE/display). Edit the weight there and the fields should be displayed in the selected order on the node view page.

Edit: In the fields screen (admin/structure/types/manage/CONTENT TYPE/fields) the weight is used for displaying the form fields in the edit/add content type form.