Instagre.at is a great new web app that shows you what’s currently popular on Instagram. Beautiful minimal design and animation blends perfectly with the Instagram theme.
UPDATE 06/11/2012: This is now possible “natively” by going to instagram.com/profile_name. Source.
UPDATE 17/04/2020: But clicking any picture opens a login popup instead of details
Yes, if you know your friend’s Instagram username, you can use Webstagram. Search for his/her username and see his/her photo stream. (E.g.: Ariana Grande’s photos.) However, note that if the respective user’s profile is set to private, you can’t see the photos. Source.
Best Answer
You can't.
Instagram by design does not allow you to do this.
They do have an API however (Docs here: http://instagram.com/developer/endpoints/media/ ) which with a bit of programming will allow you to get what you want.
As for doing it without programming, unless someone knows of a secret URL, then as far as I'm aware it's impossible.
Even doing it by web scraping (Without using the API) would require some level of programming to get the data.