Code Golf Image Downloader
Mathematica, 211 210 bytes

i=Import;FileNameTake@#~Export~i@#&amp;/@ImportString["body"/.("items"/.i["http://api.stackexchange.com/2.2/questions/"&lt;&gt;InputString[]&lt;&gt;"?site=codegolf&amp;filter=!*Lgp.gEWHA6BNP.l","JSON"])[[1]],{"HTML","ImageLinks"}]


Ungolfed:

i = Import;
FileNameTake@#~Export~i@# &amp; /@ 
 ImportString[
  "body" /. (
    "items" /. 
      i["http://api.stackexchange.com/2.2/questions/" &lt;&gt; 
        InputString[] &lt;&gt; "?site=codegolf&amp;filter=!*Lgp.gEWHA6BNP.l", 
       "JSON"]
  )[[1]], 
  {"HTML", "ImageLinks"}
 ]


It's pretty straightforward. I've set up a filter for the StackExchange API, which returns only the body of a question. The code retrieves the question information with that filter and parses it as JSON. I select the correct element (the body), and use ImportString to parse the HTML and filter out all image URLs. FileNameTake@#~Export~Import@# then downloads each of the images and stores it in the current working directory with the same file name as that in the URL.

You can find out the current working directory with Directory[].

In principle, there's a much shorter version, because ImportString can actually download all the files right away, instead of just giving me the URLs. But then I lose information about the original file type (since they are converted to Image objects upon download), so I can only save them all as the same type (PNG, say).