Since you have a criminal record, you cannot travel to the U.S. under the visa waiver program. You may or may not qualify for a visa. See sites below.
For up to six months of visit, you do not need a visa to enter USA. I am from Japan and been living in USA for 35 years now, whenever my folks come for a visit to USA, they do not have any visa. They of course visit for only 2 weeks.