You answered a question about your eligibility to travel incorrectly and were denied authorization as a result you can contact DHS for a review of your eligibility.  The U.S. Embassy cannot fix or correct any errors on your behalf.

If you believe that your ESTA was denied in error and would like to seek redress from DHS you may do so using the DHS Traveler Redress Inquiry Program (TRIP).

Any traveler whose ESTA was not authorized will require a visa for all future travel to the United States.  Instructions on how to apply are here.  Please note that all visa applicants must demonstrate that they meet the qualifications for the visa category.  Visa applications made as a result of an ESTA denial may require additional administrative processing, extending visa processing times before a decision can be made.

If you have a criminal record you will mostly likely need a valid visa to enter the United States. Some crimes will require you to additionally apply for a Waiver of Inadmissibility.

Travelers who have been refused a visa under the provisions of Section 221(g) or 214(b) of the Immigration and Nationality Act are not prohibited from traveling visa free under the Visa Waiver Program and submitting a registration under the Electronic System For Travel Authorization.  The decision to grant permission to travel will be made on a case by case basis.

Each VWP traveler must have an approved ESTA for the passport he or she plans to use for travel to the United States. If travelers become citizens of a visa waiver program country and acquire a new passport from that country, they must submit a new ESTA application for their new passport.

A list of all FAQs can be found here:
http://www.cbp.gov/travel/international-visitors/frequently-asked-questions-about-visa-waiver-program-vwp-and-electronic-system-travel