People at an organization
Linking contacts to a company, the roles they hold, and which one an invoice or project uses.
An organization's people are contacts linked to it.
Roles
A contact holds a role at an organization: the person who signs, the person you work with day to day, the person in accounts payable. The role is what tells you who to email about what six months later.
One person can hold a role at more than one organization, which happens more than you would think with contractors and agencies.
The primary contact on a project
A project points at one contact as its primary. That person must belong to one of the organizations on the project, which is Steerd stopping you from pointing a project at somebody unrelated to the client.
Who an invoice goes to
An invoice is addressed to the organization, not to a person. Contacts do not appear on the document.
That surprises people who expect "attention: Anna" on the invoice. If you need that, the client's own reference field is the place for it: it is printed and is exactly what a purchase-order number or a cost center is for.
Removing a person
Removing a contact from an organization breaks the link, not the contact. Their record and history stay; they simply no longer belong to that company, which is what happens when someone changes job.