For it's whole life, Snapchat has been strictly a mobile-only app. Even an iPad version of the app isn't available. Now, though, Snapchat can be (almost) completely accessed on your computer's web browser.
By going to web.snapchat.com, you get access to everything you see in the image above. By clicking the camera icon in the center you can take a Snap, and make simple edits like adding text. On the left you have a sidebar showing all of your recent contacts. You can directly send them Snaps by clicking the camera icon on the right.

By clicking on someone's profile, you enter a chat with them. It's very locked down, as if you click out of the window it hides the conversation. It also prevents you from taking screenshots, hiding the conversation if you hit command + shift (on Mac at least). This is unlike the mobile app, which just notifies the other party when you take a screenshot without any attempt to prevent it. I got the screenshot above (just for the purpose of writing this) by screen recording first, which wasn't detected.
You can also make video and audio calls on Snapchat on Web, with adequate video quality. FaceTime still reigns supreme in the video-chat world, so there isn't too much reason to use this web-based client. If you only have the contact on Snapchat this is very functional, though.
I can't see this taking off in popularity anytime soon, but it's a functional alternative to the mobile version of Snapchat if you don't want to go to your phone so frequently. There's no lack of functionality, but no gain too. Take this site as you will, since who knows how long it'll be functional for.