Since a participant can attend more than one event and more than one particpant will attend a given event, you need a third table:
Participants:: ParticpantID = Registration:: ParticpantID
Events::EventID = Registration::EventID
You can define additional fields in Registration to document the status of a participant's registration. (Have they paid yet, did they actually attend, etc.)
Place a portal to Registration on the Participants layout and add fields from Events to the portal as needed and you have a list of all events for which a participant has registered. Place a portal to Registration on the Events layout with fields from Participants added and you have a list of participants for that event.
Thanks! Creating the linking table sorted out my difficulty.