Ok, now I understand.
Categories are tags which are attached to articles and images. They can mark them as being about something or someone.Have a look at this picture here . Hover your mouse over the picture and click the little "i" in the bottom right hand corner. This opens the file, at the bottom of the picture you can see that it's in the categories Category:Images of Dolores Abernathy, Category:Images of characters, and Category:Images of Fan Art. You can add it to other categories, tag the image with other categories, by clicking the Add Category button, by the list of categories.
Having the image tagged with "Category:Images of Dolores" makes it appear in the automatic gallery on Dolores' page - this is why that gallery doesn't have an Add button.
Hope this helps, just carry on adding pics and dropping me a line if it doesn't :)