A report in Recode today suggests that Apple is planning to bring Apple Pay to websites later this year, which will let you buy stuff online with your iPhone. An incremental change to a marginally popular payment service doesn't sound exciting, but it's arguably the biggest payment move Apple has ever made.
Apple Pay online would be reasonably simple: on any participating website, hit 'pay with Apple Pay', and you'd get the same Touch ID prompt you do when trying to buy apps or music on a Touch ID-enabled iPhone or iPad.
There's a whole bunch of reasons why Apple Pay hasn't become universally accepted, but one thing always stands out when people are complaining about paying by phone: the time delay. It takes longer to pull out your phone and press the right buttons than it does to just swipe a card, so once the novelty wears off, people mostly forget that Apple Pay exists.
Online is a whole different thing. Every time you buy something from a new site, you're required to fill in all your credit card info and billing address by hand, which is a pain if you're trying to buy a five-pack of underwear. Anecdotally, the convenience of one-click purchases is the only reason I spend all my money on Amazon; that ease of purchase spread over the internet makes my bank account tremble.
Of course, companies have tried to make a universal payment system before, and PayPal has even succeeded somewhat. But even if a website supports PayPal, you have to log in (which takes time), using a passcode (which is unsecure). Apple Pay backed by TouchID would be faster, more secure, and have the added benefit that PayPal wouldn't get the chance to be a jerk.