Around June of 2010, Apple refreshed the Mobile Me website. Part of that refresh was an updated webmail. In Apple’s own words:
As your inbox grows, MobileMe webmail gives you new ways to manage it — including mail rules that organize incoming messages across all your devices, single-click archiving for messages you aren’t ready to delete, and improved junk mail filtering.
Archiving is a great way for getting email out of your inbox but not deleting them. It allows for fast processing of email. Below you will find a guide for getting started with Archiving using Mobile Me on the web and your Mac.
There is no native way to enable this on iPhone as of iOS 4.3.
As of this writing there isn’t a way to accomplish the Archive functionality natively in Mail.app via preferences. However, it can be done.
Interestingly enough, if you have a gmail account, you can natively Archive email on an iPhone using the built in Mail iPhone app.
As for the Archive extension for Mail, it works with Gmail too.