I'd like to use Instagram's API to display a gallery of just my own photos on a webpage. Is this possible?
Take a look here: http://instagram.com/developer/endpoints/users/
Most of the endpoints require users to be authenticated. You can retrieve tagged pictures and popular pictures without authentication. In order to display your own, you would need a user to be logged in with Instagram.
EDIT: Check this out: http://www.blueprintinteractive.com/blog/how-instagram-api-fancybox-simplified