The US still operates it's Visa waiver program (although eventually it will stop it, it has tried to before but it keeps getting postponed).
In a nutshell if you have a UK passport which has either Biometric data on it (a new one) or one of the older ones with machine readable strips then you are eligble to get into the US without a Visa. There are other conditions though (purpose of visit, if you have a return ticket to get out again, duration of stay etc). The links below will help :).
Wizzard to see if your eligble for the Visa Waiver program:
US Embassy in London:
When you say European passport, it needs to be from one of the following countries (not all Europe I know) or else you will need a visa:
Citizens of the following countries: Andorra, Australia, Austria, Belgium Brunei, Denmark, Finland, France, Germany, Iceland, Ireland, Italy, Japan, Liechtenstein, Luxembourg, Monaco, the Netherlands, New Zealand, Norway, Portugal, San Marino, Singapore, Slovenia, Spain, Sweden, Switzerland, and the United Kingdom may travel visa free under the Visa Waiver Program.
Again there are other conditions, so check you are eligible with the above links.