There are three types of identities supported through identity management in Office 365 sections: Cloud IDs, Cloud IDs with directory synchronization and Federated IDs. The type of identity approach you take will affect aspects of the user experience, administrative requirements, deployment considerations and capabilities using Office 365.
Built for small organizations that may not use Active Directory.
Built for larger organizations with on-premise Active Directory.
Read more about directory synchronization
Built for larger organizations with on-premise Active Directory and require Single Sign-on.
Federated IDs (Federated Identity) is a more sophisticated approach for larger organizations. In companies with Federated Identity set up, users can sign into Office 365 services using their Active Directory credentials. The corporate Active Directory authenticates the users, and stores and controls the password policy. With federated Identity, credentials are authenticated by on premises.
Read more about Active Directory Federation Services (AD FS) 2.0
Source: Vijay Kumar