As Dzug2 indicates, you qualify for visa-free travel if you've never been arrested or convicted of a criminal offence. If so, you can apply for an ESTA online. (NB: There are lots of rogue sites which add their own 'administration fee' onto what the US authorities charge. Use Dzug2's link and nothing else).
There is no minimum passport validity for travel to the USA (as long as you hold a UK passport). Travellers from most countries need 6 months validity on their passport but the rule is waived for Brits. Your passport only needs to be valid throughout your stay.
If you don't qualify for an ESTA you have to apply for a visa. That involves attending an interview at the US Embassy in London. The waiting time for an interview is usually around 2 weeks. If the applicant has not committed an offence involving 'moral turpitude', they might only have to wait around 5 days from then to get their visa (if one is granted). Any offence involving moral turpitude (e.g. shoplifting) means that the application has to be referred to Washington, with the process normally taking a minimum of 5 months (and often much longer). The longest wait I've seen reported on this website was one of 15 months, from someone who was eventually refused a visa because of two convictions of driving without insurance:
http://london.usembassy.gov/b2.html
Chris