Social Folders (free up to 2,000 files downloaded) — Sync photos from Facebook, Flickr, Photobucket, Google Docs, Twitter, Instagram, Picasa, and more to a folder on your desktop. Use SocialFolder to move files easily from one website to another, or send and receive files to/from your friends.
Pick & Zip (free) — Download albums, photos tagged of you, and videos from Facebook through this free software. Pick which media you'd like to back up, and the software will compress the video and images into a .zip file or .pdf.
Facebook2Zip (free) — Log in to Facebook; then, in addition to your own photos, select which friends you'd like to download albums from, choose albums, and Facebook2Zip will download the photos into a .zip file.
IFTTT Recipe (free) — This won't download the photos you've already been tagged in, but it will save photos you get tagged in from here on out. Every time an image is linked to your Facebook account, IFTTT (If This Then That) will add that file to Dropbox. Think this recipe is cool? There are plenty of other nifty automated commands that IFTTT can do for you.
Move Your Photos (free) — This Chrome browser extension makes migrating to Google+ easy, by exporting Facebook photos to Picasa. Share photos with any of your Google+ circles instantly from Picasa or download them to another backup source.
PhotoLive (free) — Install this browser extension into Chrome, Firefox, Safari, or Opera and download your own photos or those of friends. We like PhotoLive because the extension doesn't have any Facebook log-in buttons — the extension doesn't keep track or store any user activity.