iOS: Instagram doesn't let you save any of the images you see on the app. Likes is a neat and free solution to download any photo and video to your iPhone or iPad's Photos gallery.
All you have to do is first Like a post on Instagram, which then shows up in the Likes app. From these favourites, tap any that you want to save, and the app will download them to your gallery. If you want to un-Like them after downloading, long-press the thumbnail to be taken to the post in your Instagram app.
Likes is only available for iOS at the moment. Android users can try EasyDownloader to grab images.