While this is a web-app and not a desktop app (like many of Google's app), that fact makes it easier for third-party developers to create their own Android Messages experiences with electron based apps and even UWP web wrappers which we can expect to start to pop up here and there over the coming months. Depending on how many messages you send with pictures and other large files, it could be a deal-breaker in some situations. You'll need to open Messages for web on your computer, then scan the QR code there with your phone.
So today, as part of the transition, Google is bringing over some of the best features from the now-abandoned Allo chat app over to Android Messages, including the ability to send texts from your computer, a built-in GIF searching tool, smart replies, and previews for links. It's likely that Google will soon update the web app to support additional features. Messages for Web works when your smartphone is connected either via mobile data or WiFi.
While this is nearly certainly the biggest and most exciting changes for Messages, Google is also pushing a number of other new features to the app, including Smart Replies, GIF search and link previews.
Are you looking forward to texting in your browser? For starters, it has a smart reply feature. You can tap the plus button on the left side of your compose bar to search for GIFs to add to your chat.
Meanwhile, Google mentions in a support page, "Your latest conversation threads, contacts, and other settings will be encrypted and cached on your browser".