Since I'm a heavy Gmail user, my app of choice is Inbox for Gmail, and it's worked well at keeping my inbox manageable.
A very highly rated app. It's biggest "trick" is that it integrates with over 55 other apps. For example:
Other apps it integrates with include: 1Password, Asana, AppigoTodo, 2 Days, 2Do, Appigo Todo, Asana, Box, Chrome, Clear, Dolphin Browser, DashPlus, Drafts, Droplr, Evernote, Facebook, Fantastical, Finish, Google Maps, Google Translate, HockeyApp, iCabMobile, Instapaper, Message, Mercury, Omnifocus, Pinner, Pinbook, Pinswift, Pushpin, Pocket, Pocket Informant, Readability, Reminder, Safari, Skype, The Hit List, Things, Trello, Todoist, Toodledo, Twitter, Wunderlist.
Boxer claims to be the fastest, most innovative email, calendar, and contacts app available for iPhone, and I'm not really about to argue with that. It offers support for Gmail, Outlook, Exchange (2007 and up), Yahoo, Hotmail, iCloud, & IMAP accounts.
Very powerful app.
If your email is littered with newsletters that you want to remain subscribed to but don't have the time to organize, then this could be the email client for you. Spark automatically determines if an email is a personal message, a newsletter, or just an automated notification.
Triage isn't a true email client, instead it "is for people who struggle to stay on top of their inbox. It doesn't try to replace your desktop mail client, but lets you use your downtime to quickly remove the noise and stress."
When you sign in to your mail account, your new messages will appear as a stack of cards. Simply flick messages up to archive, or flick them down to keep in your inbox for later. You can also tap a message to view its full contents, and compose a short reply.
Triage supports Gmail, Yahoo! Mail, and iCloud Mail accounts, and most email services that support IMAP.
Who said Microsoft can't develop great apps for iOS? Outlook is one of the best - if not the best - mail apps for iOS out there. And don't let the Outlook name fool you, this app has nothing to do with the desktop application that man remember (usually with dread and hated).
Outlook for iOS works with Microsoft Exchange, Office 365, Outlook.com (including Hotmail and MSN), Gmail, Yahoo Mail, and iCloud.
Seamless inbox management
Easy access to calendar and files
Work on the go