As has been stated,
minor motoring offences (such as speeding tickets) don't prohibit entry to the USA under the Visa Waiver Program. However, anyone who has ever been arrested (even if they were totally innocent) or convicted of an offence by a court (including motoring offences which result in a court appearance) is obliged to apply for a visa:
http://www.usembassy.org.uk/cons_new/visa/niv/ arrests_and_convictions.html
It has been frequently pointed out, here on AB, that the US immigration authorities don't have direct access to UK criminal records. Many people simply lie when they fill in the form on the plane and get in without any problems. (Anyone, without a visa, declaring an arrest or conviction on the form would be denied entry. The immigration authorities have no discretion under such circumstances).
However, the UK does pass some information on to the US authorities. Anyone with a conviction for a serious offence (or possibly any offence involving drugs, firearms, explosives or sexual misconduct) might be barred from entry if they did not possess a visa.
The visa application process, for people with criminal convictions, is extremely lengthy. (Allow at least 5 months, possibly longer). For much of the time that the process is underway, the applicant will effectively be barred from any international travel because their passport will be held at the US embassy. Details of the visa application process are in my posts, here:
http://www.theanswerbank.co.uk/Travel/Question 289769.html
Chris