If I live in the U.K, what do I need in order to vacation in the U.S?

Here's the US State Department page on tourism

https://travel.state.gov/content/travel/en/us-visa...

The UK is a visa waiver country so you shouldn't need a visa to visit the US, as long as it's for less than 90 days.

Bring lots of money, or some high-limit credit cards.

You'd need an ESTA

It doesn't matter where you "live". What's matters is where you are a citizen and how long you will be in the U.S. If you are a citizen of Canada and will be in the U.S. less than 6 months or you are a citizen of the UK or most other western European countries and you will be in the U.S. less than 3 months, then you only need a passport and nothing aside from a passport. If you will be in the U.S. longer than 3 months and are not a citizen of Canada or of the U.S., then you need a visa. If you are a citizen of most third world countries or you will be working while in the U.S., then you need a visa regardless of the time that you will be in the U.S.

Assuming you are a British citizen you need:

Passport.

Return air ticket or air ticket exiting North America before the 90th day.

ESTA:  https://esta.cbp.dhs.gov/

Funds sufficient for the duration of your visit. 

Unless your home medical insurance covers you while abroad (unlikely) you need to get travel medical insurance which covers medical transportation to home if needed.  It is cheap and sold by airlines & travel agencies.  Please don't skimp on this as a silly accident can turn into a crisis if you don't have the insurance. 

You need your passport, ESTA, and proof of return or onward travel.

A foreign national traveling to the United States for tourism needs a visitor visa (B-2) unless qualifying for entry under the Visa Waiver Program.

https://travel.state.gov/content/travel/en/us-visa...

UK Citizens, and many other passport holders qualify for the VISA Waiver Program.

It is a lower cost and much easier and lower documentation path. Either a VISA or approved VISA waiver is required. ESTA, Electronic System for Travel Authorization is the application and hopefully approval, of VISA waiver.

https://www.cbp.gov/travel/international-visitors/...

You need a modern passport, chip embedded e-Passport

You get to this page to apply

https://esta.cbp.dhs.gov/

    You are a citizen or eligible national of a Visa Waiver Program country.    You are currently not in possession of a visitor's visa.    Your travel is for 90 days or less.    You plan to travel to the United States for business or pleasure.    You want to apply for a new authorization for one person or a group of applications for two or more persons.

The page shows you what you need to apply including a method to pay US$14 consisting of $4 to apply, and $10 more if approved. They are not responsible for banking fees and any currency conversion differences from a published number. Banking interfaces convert that to UK Pounds. ESTA is the investigation and approval of eligibility to enter and visit the USA without a formal VISA through an embassy or consulate.

The FAQ page has a lot of details.

https://esta.cbp.dhs.gov/faq?lang=en

As a resident of the UK, you can certainly understand how government agencies work both together and independently at the same time.

For the USA, there is a State Department that has the embassy and issues invitations and approvals at a country political level, a Homeland Security department to try to protect the USA from terrorists, and an Immigration and Customs at border entry that checks your Passport and permissions you received, and issues the approval to let you in for a visit of set time. ESTA is a waiver to skip the State Department and VISA process and if approved by ESTA, that gets you to the US Border where Immigration and Customs checks your PASSPORT (and online ESTA status) and stamps it and allows you in after asking questions of where you are from and where you are going and what you are bringing in with you. 

You will need a passport and a visa and a plane ticket to travel to another country. 

Since you live in the UK, you may apply for the Global Entry program which is designed to make the process once arriving in USA faster for pre-approved, low-risk travelers. 

best to look at government websites or discuss with travel agent if you are unsure. hope i helped :) 

Currently, many British citizens traveling on a valid, individual machine readable or e-passport, with a return or onward ticket, and who are staying for less than 90 days, qualify for the Visa Waiver Program and can travel visa-free with only an Electronic System for Travel Authorization (ESTA).

Tomorrow, Jan. 31st the UK leaves the EU. There might be changes, although that's highly improbable for at least a year or so.

There are some conditions, however. check them out here https://uk.usembassy.gov/visiting-united-states/

I wasn't aware that you needed anything other than a passport.