An 'Identity Provider' (somtimes abbreviated 'IdP') is a service hosted by an organisation which publishes electronic identity information for users that have a firm relationship with the organisation.
Thanks to the firm relationship, information published by the Identity Provider can be said to be trusted by the user. The relationship between the user and the Identity Provider can be compared to a bank customer having confidence in the bank's information about her economic affairs.
The user benefits from the service provided by the IdP by having the IdP publish and maintain identity information for her rather than having different other unrelated services and companies doing the same thing separately. These other services and companies may use identity information published by the IdP in lieu of maintaining it themselves and the user will not have to establish a trusted relationsship with all of them separately.
An 'Identity Federation' is a structured co-operation between different Identity Providers. In a federation, IdPs will multilaterally exchange electronic identity information, such as names and other personal data. [länk till] SwAMID is one example of an Identity Federation.