The USA have a visa waiver programme for any person holding passports from certain countries (United Kingdom passports are included) as long as that person has never been arrested and/or convicted.
If you have been arrested, you must declare it whether or not that arrest resulted in a conviction. The Rehabilitation of Offenders Act 1974 does not extend to the USA so you must declare all convictions regardless of whether they are classed as spent or not.
Most minor road traffic offences that were dealt with by way of fixed penalty (i.e. no arrest or court appearance) do not count and you will be eligible to travel under the visa waiver programme.
If you have been arrested/convicted, then you must apply for a visa from the US Embassy in London before you travel to the USA. Failure to have a visa means that you could be refused entry and returned home at your own expense. Have a look at the Visa Waiver Wizard (first link in Related Information) which only takes a minute to complete - it is just a guide for your own information, you do not have to enter personal details.
A conviction could mean that you are classed as permanently ineligible to travel to the USA, however, you may be able to apply for a waiver of permanent ineligibility from the Department of Homeland Security. This is not automatic and depends on several factors depending on the nature of the crime and when it was committed.
If you are unsure it is always better to check as you could be refused entry to the USA. In the current climate, it is highly likely that the USA authorities will be aware of your personal details before you travel to the USA.
Also, see the website in Related Information for more details.