Today I have got my invite to the new Inbox by GMail. I rush to install the software and give it a quick test. Screenshots here are in Spanish (the language set on my phone) but you can get an idea of the interface and functions. You are also given the option to use the new Inbox interface on the web.
Communication is made easier with the new compose button. It will show you immediately the altest people you have been interacting with (received or sent emails). You can also create reminders. And of course, you have always the powerful Google search button on the upper right corner to search quickly through your emails.
The idea behind Inbox is to help you keep your mail tidy and not to forget important things. You can swipe mails left to remind you later, or right, to archive them.
You can also pin any email on the mail detail view so you can later access all you pinned mails from inbox (with the pin switch on the upper part).
I think Mailbox app by Dropbox has got now a serious competitor with Inbox by GMail.
Unfortunately, no iPad verion has been released yet.
The service is still in beta. You can sign in at google.com/inbox