How to download files from the Web in the Android Browser?
Nowadays the built-in Android browser has been replaced by Chrome, which does this perfectly well.  You can long-press and save virtually any image or link, and even videos from a number of sites that couldn't be easily done on the desktop.

Answer below for older devices:

The Android browser by default can only download files that the Android system "recognizes" (i.e., there has to be a program registered to handle that file type).  It's a stupid restriction in my opinion, but you can install Download Crutch to overcome this limitation (it registers itself for every filetype).

If you're referring to images that link to something else and you want to save the something else, long-press on the image and choose "Save Link As" or similar.