Exactly what a Facebook Group or a Facebook Event actually is should hopefully, to anyone who's used Facebook, be self-explanatory. But essentially, a Group is a loose association of Facebook users. Any user can create a Group, and depending on the sign-up process specified by the owner for that Group, anyone can join. Similarly, any user can create an Event and invite other users.
Both Groups and Events have a basic set of information—name, description, and picture. Events are obviously time - and location-sensitive, although they don't automatically disappear after they've occurred, and Groups are a more long-term collection of users, with no specific venue, geographical location, or prior connection between the members.
Information about Facebook Groups in general, and a web interface that can be used to create a new Group, can be found on the Facebook.com website, at the URL: http://www.facebook.com/groups, and the same goes for Events, which are similarly available from...