Applies to users with Office 365 E3 or E5, or those with Azure Information Protection P1 added onto their account if it is not E3 or E5.
Securing Outgoing Messages
With Office 365 message encryption is now a built-in service that can be selected as an option when drafting a new email.
As an example, I'd like to send an email to an external contact, but I want to ensure that the message and/or any attachments are encrypted and secure. Perhaps it is sensitive personal information in the email or confidential business files attached to the email - in either case I want to be sure that the only person who can see the email is the person I send it to. In this case I can apply encryption to the outgoing message by going to Options > Encrypt > Encrypt-Only:

Once Encrypt-Only has been applied to an email you will see this message appear above the To: box:

You can now send the email or edit it further before sending. In either case once it has been sent it will be encrypted.
Receiving Secured Messages
Once you send the encrypted message the recipient doesn't receive the email you drafted directly, they'll receive an email which looks like this:

To access the message directly they will need to click the Read the message button. Once they do they'll be taken to this screen:

They can now login using an existing Office 365 email address (if this matches the address that the encrypted email was sent to) or by requesting a One-time passcode. In most cases the One-time passcode is the better option, especially where the recipient isn't aware of if they are on Office 365 or not.
By clicking Sign in with a One-time passcode they will be sent a code to the same email address the encrypted message was sent to, and they'll be taken to this screen:

Here they will enter the One-time passcode and can choose to keep themselves logged in for 12 hours - this is good if they will be receiving several encrypted messages from you over the course of the day. Once they have typed in the passcode they will be taken to the message:

As you can see, they can reply directly to the encrypted email, but they can't forward it on.
How This Works
As you can see sending encrypted messages ensure you maintain maximum control over who can view them and what they can do with the messages.
Essentially the email never left your company's Office 365 servers, the only thing the recipient receives is a link to access the email contents, and even this is highly controlled and managed.
Tip
One thing I would highly recommend to make it easier is to add the encryption option to the Quick Access Toolbar (the icons in the top-left of the New Email window). By default, the toolbar will look like this:

If you click on the downward arrow and choose the More Commands... option:

This will take you to this creen, where you can customise the Quick Access Toolbar:

Firstly, select the Options Tab from the list of where to Choose commands from.
Next, find the Encrypt option that has no arrow next to it, this should be the second out of three - you can see one has a right arrow and one has a down arrow.
Select the Encrypt option and click Add >>. Click OK to close the customisation screen.
Now the Quick Access Toolbar has the Encrypt option shown:

With this available you can quickly encrypt a message without needing to go to options and selecting Encrypt-Only from the encrypt option.

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