There are may third party web applications as well as installable softwares to do download individual or complete backup of Instagram images. I have tried many of them and here are the best, easiest and free ways to get your photos into your control.
If you want to download photos from your own profile that you have uploaded, this is the best way. Just validate access of the Instarchive to your Instagram account and it's just a matter of few clicks.
You will get the ZIP file for downloading packed with all the images of your own.
Instaport is also a simple online service, but with many features. You can download all the images of your own as well as the photos of others that you have liked. You can define period of the photo upload, like from and to date, last 10 photos, photos with specific tags or all the photos.
The extra feature is Instaport also allow you to download Videos, though it is still in beta.
If you are using Windows 8 computer or tablet, this is a good Instagram client that can along with basic features can save the photos. This is not a dedicated photo downloading application, but you can save the photos individually one by one to local memory. Just select any image in your photo stream and tap on 'save' button.
If you want the Android application to download all your Instagram photos, Pirctarine can do it for you. Pictarine also have the web based service that you can use on your computer. The Pictarine Android application allows you to view all the photos from various social networks like Facebook, Twitter, Tumblr, Instagram, Flickr, Google Plus, Photobucket, Dropbox and many other services at one place. You can view, comment, like the photos. It's actually a client for multiple social networks specially designed for photos in mind.
This is a free software for Windows PC. You have to enter the username of the Instagram profile and select the photos that you want to download. You can download photos from any other username also, there is no need to validate the account.
Conclusion: Instaport is the best and easiest way to download your Instagram photos with more options.