Mailbox is a replacement mail app available for both iPhone and iPad. Originally owned by Orchestra, Mailbox is now part of the Dropbox family of products. Mailbox was one of the first mail replacement apps that took a task-centric approach at handling and dealing with emails. Swipe messages to mark them for later, archive them, trash them, or add them to lists. The goal of Mailbox is to get you to inbox zero and keep you there.
Traditional email clients leave messages in your inbox until you either file them away or delete them. Mailbox can temporarily remove them from your inbox until you need to act on them. This way you can focus on the tasks you need to and save the ones that don't need your immediate attention for later. Mailbox's new Auto-Swipe feature lets you choose how to act on frequent emails. For example, if you'd always like to file emails from your boss to a specific list, just create an auto-swipe and Mailbox will abide by your swipe preferences.
If you use Mailbox on multiple devices, Dropbox can sync all of your accounts and settings so whenever you pick up a new device or need to re-download Mailbox, you'll never have to enter your accounts again. Just authorize Mailbox with your Dropbox account and you're good to go. The Mac version of Mailbox is currently in private beta and should hit later this year.
Since Mailbox's initial introduction in 2013, countless replacement mail apps have cropped up in the App Store offering similar functionality. If you don't like the built-in Mail app you've got a lot of options to choose from nowadays.
You can download Mailbox for both iPhone and iPad from the App Store for free.