Google just announced a pretty major update to Gmail, and you should now be able to give it a go in the browser, if you so desire. As our pal Paul O'Brien from MoDaCo points out, all you have to do to enable the new tabbed organization is go to the gear icon on the right of the screen and choose "Configure inbox." (If you're using a wrapper like Mailplane, you might need to do this from a traditional browser first, then restart the app.)
Next you'll be asked which tabs you want to use -- primary, social and promotions are checked by default; updates and forums are other options. Once enabled, you'll find e-mails sorted among the tabs. To move e-mails between tabs, just click and drag. (You'll also be asked if you want to continue to get that particular kind of e-mail in the new tab.)
Still no sign of the new Gmail app for Android, which also will bring a new sorting experience.