When you share a photo from Flickr, it gives you four choices:



It's important to understand that none of those choices link straight to the raw JPEG (or other format file). The choices are, in order:

• Share - generates a link to the Flickr page you are looking at, suitable for use as a hyperlink in text.
• Embed - generates a snippet of HTML you can drop into a web page, if you are creating content at that level:
  
  HTML Code:

  <a data-flickr-embed="true"  href="https://www.flickr.com/photos/49705339@N00/36841684490/in/dateposted-public/" title="share"><img src="https://farm5.staticflickr.com/4349/36841684490_4e376f05da_z.jpg" width="555" height="346" alt="share"></a><script async src="//embedr.flickr.com/assets/client-code.js" charset="utf-8"></script>

• Email - brings up a little pane where you can compose an email to people containing (I presume) a link to that page.
• BBCode - generates a snippet of BBCode suitable for inserting into forum posts like this
  share by smontanaro, on Flickr

For posting here, you will almost always want the BBCode option.

You can extract the raw JPEG image URL from either the Embed or BBCode options. Here I've messed up the BBCode slightly so it doesn't actually do what it is supposed to do (raw HTML is never rendered), and highlighted the raw image URL buried in them.

BBCode:

[ url=https://flic.kr/p/Y8zgfJ][ img]https://farm5.staticflickr.com/4349/36841684490_4e376f05da_z.jpg[ /img][ /url][ url=https://flic.kr/p/Y8zgfJ]share[ /url] by [ url=https://www.flickr.com/photos/49705339@N00/]smontanaro[ /url], on Flickr

Raw HTML:

<a data-flickr-embed="true" href="https://www.flickr.com/photos/49705339@N00/36841684490/in/dateposted-public/" title="share"><img src="https://farm5.staticflickr.com/4349/36841684490_4e376f05da_z.jpg" width="555" height="346" alt="share"></a><script async src="//embedr.flickr.com/assets/client-code.js" charset="utf-8"></script>