In a nutshell, a franchise typically involves:
1. An upfront fee that you pay to become involved and receive initial training.
2. A common brand name that all units operate under
3. An ongoing relationship that involves support for your activities and some sort of continuing payment to the company (in the form of royalties or required product purchases or something like that).
If you have all of these factors, it is definitely a franchise and the seller is required to provide you with a FDD document as mandated by the FTC.
If you are missing one or more of these factors, but there is an upfront fee to get involved, it is probably a business opportunity as the term is defined under the law. This type of arrangement also requires mandatory disclosure under the laws of a number of states. Though the disclosures vary to some degree, they are typically significantly less than a franchise is required to provide.
If there is no required fees that have to be paid, either upfront or after joining, then the business opportunity is typically not considered a franchise or a business opportunity under the terms of the laws requiring disclosure. It is simply an opportunity you may wish to pursue.
All three of these forms of opportunities can be considered an affiliation because there is a common denominator of some sort for every person that gets involved in the opportunity. The term affiliation does not connote any required disclosure but is simply a descriptive term used to generally describe the relationship.