Not too long ago, Twitter released 'Twitter Lite,' a progressive web app version of the official Twitter app. It can do almost everything the native app can, including notifications, but it uses less data and runs entirely in a web browser. Now Instagram has done something similar, with the new web app offering much of the native app's functionality.
Before this, if you visited instagram.com in your browser, you could really only look at your timeline and your profile (plus other people's profiles). Now, the web app looks very similar to the native app, and has new Search and Activity tabs.
You can even upload pictures, but you can't tag users, add a location, or share the picture to other social networks at the same time. You also can't receive notifications through the app, send and receive direct messages, or use Instagram Stories.
Still, if you want to try it out, just visit instagram.com in your mobile browser of choice. OMG! Ubuntu! also made a guide to turn the web app into a desktop application with Chrome (the guide is specifically for Ubuntu Linux, but it works on any OS with Chrome).