I'm trying to keep record of how many times an image has been downloaded.
Is it possible to detect if a client manually saves an image through their browser?
Only from server-side logs. One way that this is done is for a server-side script (such as PHP) to scrape through server access logs, looking for all occurances of what you're interested in.
No, you can not determine if the user has saved an image from the web page.
One possible way to obtain some numbers though is to provide a low-quality version of the image on the web page, but allow them to request a link to be sent to their email for a high-quality image. That way you can keep track of such requests.