The HTML code to embed a photo in a web page that you used to find on the All Sizes page is now only in the Share menu on the photo, to the left of the More menu. We now also include BBCode! (standard message board code)
There are a few cases where the code may not be available: 1) If the photo is restricted. 2) If that person has turned off "Share this" or access to original files.
The direct link to a photo file is no longer shown on the page. Per the Flickr Community Guidelines "pages on other websites that display content hosted on flickr.com must provide a link from each photo or video back to its page on Flickr." Linking directly to the photo file doesn't do this.