A criminal conviction is a public record yet -- unless you have FBI clearance -- there is no comprehensive search available to find out if a person has been convicted for committing a crime anywhere in the U.S. Instead, you can search the available online sources and hire services to check the records of other courts where an individual lived.
Some online sources to consider:
(a) A Federal docket sheet search. Use the PACER Case Locator or a Pacer-derivative database (see "Docket Sheets") to find criminal convictions from the U.S. District Courts.
(b) CriminalSearches claims to search the criminal records of all 50 states in the U.S. plus the District of Columbia.
(b) Other free and low-cost internet sites. Many state and/or county conviction and/or incarceration records are available over the Internet. You can find links to most of these sites through SearchSystems.net and/or BRBPub.com.
(c) The National Sex Offender Registry. Also the relevant state sex offender registries.
(d) A commercial database search. TLO, Accurint, TLO, FindOutTheTruth, Rapsheets, Lexis (FINDER;CRIMNL) and Westlaw (CRIM-ALL), have state-wide criminal history records for most states as well as inmate records for a few and criminal court records for a few more.
If you are looking for the police report for a particular incident (especially auto accidents), PoliceReports.us sells them for many states. For more, see the Criminal Law entry.
Note: If you need a service to check courthouse records, Lynn Peterson has recommended Accurate Background, although there are many other good services around the country too.