How can I let registered users upload images using a form in Drupal 7?
You need the field permission module. Everything is already contained in Drupal core.


Create a content type.
Add an image field and allow multiple instances of this field.
With help of the field permission module make sure registered users have the right to edit your image field.
Configure ImageCache to output the images the way you want.


As an add on you might have a look at views, wich allows you to output many nodes with their images, sorted and/or filtered the way you want.