She attended the fund-raiser 'at the invitation of' the committee chairperson. [=she attended because she was invited by the committee chairperson]
You have a 'standing/open invitation' [=you are always welcome; you are always invited] to visit us any time.
The event is 'by invitation only'. [=only people who have been invited to the event can attend it]
(..) The fat guy's still with him. They're just talking. - What are we doing here? - Protecting people. - Nobody asked us. - You need an invitation? I'd like one, yes. We keep sneaking out to do this, and... - You remember Gazerbeam? - There was something in the paper. (..)2004 The Incredibles