what you refer to as ‘widgets’ (and they do indeed look exactly as widgets) are hardcoded placeholders in the sidebar.
in most themes, when you drag your first widget (in admin-appearance-widgets) into the sidebar, it removes the placeholders.
for affiliate link images, try to use textwidgets and place the image/link code into them.
general structure of a linked image html:
<a href="http://affiliatelinkurl" title=""><img src="http://absolute/path/to/image/image.jpg" /></a>