ShazzleMail is a free web service which allows you to send private emails via a direct connection between the sender and receiver. Your messages are never stored in the cloud, where they can be accessed and mined by the company: no-one else sees them at all.
To get started you must download the iOS or Android app and set up a new ShazzleMail account (email@example.com). This is quick and easy, and works much like creating any other web account: provide your name, another email address and you're done.
Communications between ShazzleMail users are very straightforward. You create a message on your mobile device, Mac or PC, perhaps attach a file, and send it to a ShazzleMail.com address. As long as your friend is online, the two clients establish a direct connection, and your message is transferred.
Most of the people you know won't have ShazzleMail.com addresses, of course, but that's not a problem. Send a message from ShazzleMail to a regular email account, and the receiver will get a message with a "click to view" link. Clicking this establishes an encrypted connection between them and your iOS or Android device, so effectively they're downloading it directly from you.
Similarly, someone without a ShazzleMail.com account can send messages to your address.