How do I use Instagram's API to display a gallery of my own photos?
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
Method 1 - Use Instagram API
No you don't always need the Access Token. You can also use the Client ID like in this API request - https://api.instagram.com/v1/tags/nofilter/media/recent?client_id=CLIENT-ID
User Photos Endpoint -
https://api.instagram.com/v1/users/{user-id}/media/recent/?client_id=CLIENT-ID
You will need to replace your User ID in the above URL. The output will be a JSON response so you will have to display the photos yourself.
Method 2 - Use a Plugin
You can directly use a plugin that will embed your Photos from your Instagram Account onto your website. A good Example would be - InstaMax plugin which will display all your photos, videos with likes and comments.
A small Demo of the plugin is present here - http://demos.codehandling.com/instamax_demo/instamax_live_edit.html
Update 2016
Method 1 will not work after June 2016 due to changes in Instagram API. You need to have you app approved from Instagram and then generate an Access Token.
Generating the Token is the easy part. You can simple use the below API from your browser -
https://api.instagram.com/oauth/authorize/?client_id=CLIENT-ID&redirect_uri=REDIRECT-URI&response_type=token
Frankly, the hardest part is getting your APP approved from Instagram. It took me a month and several rejections to understand what their platform policies are and how to send the request for approval.
I would recommend you to use a plugin that already has an approved App so that you can use their App to generate Tokens for their plugin.