I use twitter. Some people\'s tweets contain photos and I want to save them.
I checked ifttt, where twitter is not a trigger. Thus, ifttt cannot help me do it.
Greasemonkey cannot do this, because ordinary javascript is forbidden to do this (for security reasons), and the Greasemonkey API does not expose a method to write files (and never will).
Here are four alternatives:
Update: Switch to Tampermonkey, which you should do anyway. Then you can use GM_download as user136036 said in his answer.
or
Install and use the excellent DownThemAll add-on (Update: Firefox 57 withdrew support for this kind of extension). It still requires one click, but that's better than always grabbing a file willy nilly, in most cases anyway.
or
Write your own addon extension. See this (now obsolete) answer for file-writing code from one of the top gurus of FF add-ons. But "new" style extensions can still do this.
or
Use XAMPP (or similar) to run a web server on your machine. You will then have to write a web application that excepts incoming image data (or just the image URL) and saves the image to disk.