Both American and UK citizens can travel to each other's country thanks to Visa waiver program. If I understand correctly... The agreement is between USA and the whole of Europe so what that program allows is that you can visit USA/UK without obtaining a visa if you are on vacation as a tourist but you would require a business or work visa if your purpose of travel is indifferent.
Passport holders of any first world nation can travel to roughly 160-175 countries without the need to obtain a visa bit they cannot work in most of the countries. After the freedom of movement agreement within Europe.. now European citizens can travel, work, live in any other EU country without any restriction but this rule only applies within Europe. If an American, Canadian or Austrian citizen wishes to come and reside in any of the Eh countries + Switzerland and Norway... They would need to see permit for that
Hope I was clear enough in explaining few basic rules of visa policy?
