The latest update to Pushbullet for Android enables full SMS support on your computer, through the same company's assorted Web interface or browser extension.
On your computer, Pushbullet has been showing you individual texts as they arrive on your phone even up until this point, while also letting you reply to them on the spot.
But from now on you get pretty much a full messaging experience on your computer - it's like your SMS app moved from your phone. You see all your conversations, and all the messages inside each of those.
You can even be in multiple conversations at the same time - clicking the arrow to the right of a person's name opens a chat window. Sending texts happens when you hit Enter after writing something.
Everything is synced to your phone, as you'd expect. And obviously the messages themselves are sent from your handset, so the people you communicate with won't know you're actually using a desktop app (or a browser extension).