Instagram is a way of sharing photos, but what's unique is that it has filters that it adds on the photos to give them unique looks, looks like 1960s photo effect and different effects of that type. Its usually automatically applied and it gives a unique look to all photos.
I do not understand what use this Instantgram serves. As far as I am aware, if you want to put a photo on Facebook, Twitter or other places, you can do this straight away with no middle man. If you want to store photos somewhere, you can use Flickr.
It is mainly because of the surge photos taken with a camera-phone. This service allows you to make photo edits as well as other geo-tagging measures that allow the photos to be used more for social sharing. Older methods include downloading the photo then uploading it to a photo editor then downloading it again... it really is a time saving tool with an added edge to it. There is a downside to it though because the service has to earn some money it uses the geo-tagging info to target advertisers using your content. It is being contested at the moment.
I also read recently, and I can't for the life of me remember where, that if you have photographs on your FB page which have been altered with instagram, they will become of the property of FB after a certain date in January. I'm not sure if this was a windup.