Apple provides a pretty efficient mail app for iOS, but third parties are always trying to be better and some users are always looking for more. Inbox Cube is a free iPhone app (it's not universal, but will work on the iPad) that makes some improvements on the basic Apple Mail app.
Inbox Cube supports iCloud mail, AOL, Yahoo and Gmail. One feature I think is very strong: the ability to just see photo, video or document attachments in a single click. That includes attachments you have sent as well as received.
The on-screen display is attractive, and it is easy to navigate. You get notifications of new email just as fast as you would with the Apple mail client. I had them both activated and heard the mail alerts at the same time from both.
Search for content in mail is very fast. In addition to global search, you can also search by subject, text in the body of the email or the from / to fields. When emails are listed, each has a colored icon with the first letter of the senders name, which makes for a quick visual way to sort. Each letter is a different color. You can swipe through mail and attachments. You can also assign favorites for mail you want to get to easily.
I don't see any support for POP or explicit support for Outlook. So, if you are outside what is supported, beware. It should be easy to set up with most IMAP-compatible systems. Outlook does support IMAP clients, but I was unable to test it. You can have multiple accounts, for example, on Gmail or the other supported mail protocols, and view them separately.