After trying for a long while to figure it out, I still cannot convince my Android browser to save the file from a web link. If the link is a picture, holding over it and pressing "Save As" saves the picture, not the file it links to. My Android device has no multitouch (is this required to tell to browser to download?), and simply clicking on a download link takes me nowhere. Can anyone help me?
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.
Personally I usually just click the link of the file and it downloads.
As for the image: try long pressing but instead of choosing "Save As" look for "Copy link URL" or "Open in New Window" or something similar to those two menu options. "Copy link URL" should copy the URL of the file to your clipboard, then you can paste that link in the address bar and hit "Go" and it should download your file. "Open in New Window" should essentially do the same thing (i'm just trying to give you as many options as I can to try). If you click on the link it may not take you anywhere but to a blank page while the file downloads.
So to check your download list: You may have already tried this but to see if the file was downloaded you can open your browser, open the menu, then select "More", you should see "Downloads". This is where your downloaded files list is.
If this is not a solution let me know and give an example of what you are trying to accomplish and I'll try to figure it out.
I've struggled with this same issue, though in my case I wanted to save a Text file (.txt) that I had open in the Chrome browser on Android 4.4.x (Kit Kat). After reading this and other questions posted I realised that neither Chrome, the Google applications, nor the standard Android tools would allow me to simply save a Text file from a URL.
To save a Text file from a URL to the file storage on my Android device I used the File Explorer application ES File Explorer. Within ES File Explorer there's a Download Manager that allows you to download any file from a given URL to the file storage on your device.
What Android OS version are you running? And what kind of file does the image link to?
On 2.3.4 the browser offers me "Save link" and "Save image" as two options when I click on an image that is a link (e.g. on the image link to the Main Page at http://en.wikipedia.org/wiki/Web_browser)
But in earlier versions of Android it was different. E.g. in 1.5, I just see "Save image" and "View image", so there is no direct way in the default browser to save the web page which the image links to.
As Matthew notes, the file type can also matter. On my 2.3.4 phone it asked me if I wanted to save a file for which there was no handler. E.g. for an xml file it said "The content is not supported on the phone. No application can be found to open this file. Do you still want to download it?"
For earlier phones and some kinds of files you may need to install an app which can handle the file. "Download Crutch" is no longer in the market, but there seem to be others that would be suitable.