While there is no straightforward way to download other people's pictures from Flickr (though, if they have a Creative Commons license, you may be allowed to), you can easily download any pictures or videos from Flickr that you've uploaded yourself. This is handy if, for instance, you accidentally lose or delete the original copies of the files on your device, and need to replace them.
How to download photos from Flickr
Go to www.flickr.com in your web browser and log in.
Move your mouse cursor over You to make a drop-down menu appear. Click Camera Roll to view a list of your pictures, organized by the date they were taken.
Click on the photos that you wish to download in order to select them. When selected, they will be outlined in blue and have a checkmark on them. (Click a photo again to deselect it). When you've selected all of the photos that you wish to download, click Download at the bottom of the screen.
A pop-up will appear confirming the number of photos that you are about to download. Click Download ZIP.
A download prompt will appear. Make sure the button beside "Save File" is filled in (click it to do so if it isn't), and then click OK.
An explorer window will pop up, allowing you to choose where you want to save your photos. Navigate to the location that you want your files downloaded to, and then click Save
Your photos are in a .ZIP folder, which means that the files are compressed into smaller versions that take up less memory space. To get them back to regular size, right-click (or hold the "Command" button and click on an Apple computer) on the folder, and then click Extract All.
A window will appear, asking where you want to extract the files in the folder to. If you're okay with where they are right now, just click Extract. Make sure to have the check box next to "Show Extracted Files When Done" marked (if it's not, click it to mark it).
Voila! There are your downloaded Flickr photos, in all their glory!
Now you know how to download your photos (and videos) from Flickr if you ever need the original files for some reason!