The example used most often to demonstrate this is using a "Registrations" table to join a students table to a classes table so that one student record can be associated with many class records and one class record can be associated with many student records. Relating a single student and class to a record in the registrations table (for as many associations as you need) means you can put a report of all the registration records on the class showing the related students and different report on the student record showing all the related classes. Whatever data you want to appear in those reports you "push" down to the bridge table through the relationship with the parent tables.
It is a little hard to explain, but the easiest way for you to explore this may be through an app in the exchange,
Many-to-Many Relationship Example App which seems to use this classic example. In your case, replace classes with organizations, students with contacts, and I might make a bridge table called "Members" instead of registrations, but "Organization Contacts" is fine as well.
