There's no lack of apps that pop up when you search Google Play or the App Store for "Instagram download" and they all look much the same. It's hard to really recommend any of them, but going by the in-store recommended reviews, the top rated app as of this writing appears to be Video Downloader for Instagram(Opens in a new window) (free for Android) from ETM with a 4.9 star rating across almost 10.5K reviews. There doesn't appear to be anything like that in the App Store for iOS users, not that doesn't lean heavily on the "repost" paradigm.
You've now got the tools and info you need to put almost all of Instagram on your own hard drive. Use your powers only for good. Remember, there's a reason Instagram posts are generally easy to share via embedding without any downloads, because it takes away from no one. 2b1af7f3a8