When someone is convicted of a crime in a United States courtroom, it is a public event. The record of the conviction is a public record, and anyone can go to a court clerk's office and search the files for records of conviction for a certain person.
The only exception is for convictions that a judge has sealed. A convicted person may ask a judge to seal (or "expunge") a conviction if the legislature has made that an option for the crime committed. The requirements for sealing a record are usually that the offense was the defendant's first one, that it was a relatively minor transgression (such as possession of small amounts of drugs for personal use), and that the defendant successfully finished the sentence (such as probation), and did not re-offend within a specified time. People who have a sealed record for one conviction may say "No" when asked if they have been convicted of a crime. (To learn more about sealing criminal records, see Nolo's article Expungement of Criminal Records: The Basics.)
The FBI and state criminal records offices keep databases on convicted persons, but access to those databases is restricted to government personnel, such as prosecutors and police, and some employers. Until relatively recently, if you wanted to find out if someone had a criminal record, your only method would have been to visit the clerks' offices of every county in your state -- and in the country, if you wanted to be really thorough. But with the advent of the Internet and sophisticated data gathering, you can now use the services of the ubiquitous criminal background searching firms that operate online.
Criminal background search services have access to conviction databases gathered by private, for-profit companies that canvass courthouse conviction records, either the old-fashioned way (by sending people to check the files) or by trolling the Internet, picking up public notices of convictions posted by courts. The value of such databases is somewhat iffy, since they depend on accurate and timely gathering practices. Ironically, fear of identity theft keeps many posting agencies from using defendants' Social Security numbers or dates of birth; but this lack of identifying information makes mismatches very common.