If you are going to the USA from a Visa Waiver Country (Which the UK is one) and have a previous conviction, you will most likely have to apply to get a visa to enter the country. It's a little more hassle, but unless your offense was pretty serious (in the eyes of the USA), doesn't mean that you can't get in.
To be declined a visa, you would need to have committed a crime "involving moral turpitude" or related to "a controlled substance", though if the crime was committed when you were under 18 years of age or the maximum penalty of the crime is less than 1 year or you didn't serve more than 6 months.
Hope this helps.
If you have multiple convictions, regardless of the offense, you are not allowed to obtain a USA Visa, and cannot visit the country.