Portrait mode is increasingly available on smartphones, especially those with two cameras. These pics have blurred backgrounds to make the subject pop like you were using a lens with a shallow depth-of-field. Some phones do this better than others, but I think all of them do it better than Instagram, which just added "Focus" mode in the latest update.
You can pick Focus from the mode selector at the bottom of the camera UI. As soon as it's active, the app searches for a face. When it finds one, the app applies a blur filter to the background. Although, it's not always the best at determining where the background starts. You'll probably end up with a lot of soft edges. The quality of the blur filter is questionable as well. It looks "wet" blurry rather than "unfocused" blurry.
You can use Focus mode for a photo or a video. Like other modes, you simply press and hold the shutter button to capture a video. Focus mode is live in the app right now. Just make sure you're on v39 (both Android and iOS). It won't work on all Android devices, though. Instagram just says "select" phones. We're asking for clarification.