I have had a few customers ask me recently about low level details around how authentication works for Office 365 and Active Directory Federation Services (ADFS). There is a great whitepaper located here that I highly recommend that you read - http://www.microsoft.com/en-us/download/details.aspx?id=28971
I recommend reviewing sections:
Section 5.2 covers Web Client for SharePoint/OWA authentication flow.
Section 5.3 covers the Lync authentication flow.
Section 5.4 covers the Outlook authentication flow.
All are a little different because clients are active vs. passive which determines the flow of how the claim is ultimately sent to Office 365.
I currently work for Microsoft as a Sr. TSP for Office 365 and SharePoint supporting sales for US Federal. Prior to Microsoft I worked for five years at a Microsoft Gold partner called RDA as a Technical Project Manager. Over the past years I have done a lot of work with MOSS, Silverlight, K2, BizTalk, InfoPath, ASP.net, etc. I have written a Wrox book about SharePoint BPA technologies and one of the solutions was a finalist at Microsoft Worldwide Partner of the Year conference. I am a Virginia Tech alumnus completing both my undergrad and masters in Information Technology. I play ice hockey on a regular basis and I am huge Washington Capitals fan.