It doesn't matter what you use for your baseline email account—services like Gmail, Yahoo, iCloud, Outlook.com, or systems that use Exchange or IMAP or even POP3—you can and should access your messages on the go with your iPhone. It's not like email has really changed all that much in 30 years, but the experience is still totally modern when you take it on the road.
Apple includes a simple yet pretty powerful app called Mail with iOS—in fact (until iOS 10 comes out this fall) you can't even delete it. But you may not want to. Mail.app, as it's sometimes referred to keep things straight, does pretty much everything you'd want in a mail client: it sends, receives, replies, composes, and organizes messages. It throws in a few extras which we'll try to cover here.
That said, there are also some unique features to be had with certain third-party email apps, many of which first emerged in a very hyped app that no longer exists: Dropbox's Mailbox. Several current iOS email apps continue to emulate it, and there are plenty out there. There's Gmail and the separate Inbox by Gmail, as well as Yahoo Mail and Outlook, which earned our Editors' Choice.
Then there are a slew of third-party apps like Inky, Boxer Pro, Inboxcube, Spark, CloudMagic, and SaneBox (also an Editors' Choice). For more, look at The Best Mobile Email Client Apps.
Sticking with the original Mail app isn't a bad thing, however. Whatever you use, take a look at what is unique—it may help you decide on exactly what app you want serving up your inbox. Or better yet... run all the email apps you like and pick the right one for the right job.