The phrase Best Effort Internet refers to the model of the Internet that does not differentiate between "levels" of content providers. All web authors, large and small, enjoy the same ability to produce content or services that can, via the Internet reach an audience / customer base. The phrase infers that success on the Internet will result from those putting forth the best effort, and not from "artificial" barriers constructed by telecommunications carriers who manipulate service via bandwidth control.
Advocates for the Network Neutrality principle believe the "best effort" model currentlty (June 2006) in practice, should remain the dominant paradigm of Internet use.