As has been indicated, an ESTA will suffice for many people.
However if you're ever been arrested (even if totally innocent) or convicted of any offence you're supposed to apply for a visa.
If you've actually been convicted of an offence you're ineligible for a visa. If that offence was one of 'moral turpitude', which could simply be nicking a Mars bar from a shop, US law prohibits the authorities from issuing you with a visa. You can appeal against but to do so typically takes over a year.