Foreign nationals with one or more criminal convictions in their past are typically ineligible to receive an immigrant visa to enter the U.S. (non-immigrants are likewise barred, although the bar is not quite as broad as it is for immigrant visas).
There are some exceptions to the bar, however:

• The offense is not a crime of “moral turpitude” (a very slippery concept that some courts have equated with a fundamental “baseness”, “depravity” or “vileness”, but for which no clear definition exists)
• The offense was “purely political”
• The foreign national has (1) committed only one crime involving moral turpitude at any time; AND (2) “was not sentenced to a term of imprisonment in excess of six months (regardless of the extent to which the sentence was ultimately executed) AND (3) the conviction carries a maximum possible sentence of one year or less.

Drug traffickers are ineligible for a visa, even if there has been no conviction, as long as the consular or immigration officer knows or has reason to believe that the visa applicant has been involved in trafficking.

A person coming to the U.S. to engage in prostitution, or who has engaged in prostitution within ten years of their application for entry, is inadmissible, even if there was no criminal conviction.

An amnesty or parole does not remove the crime from calculation of the bar. Such crimes are treated for U.S. immigration purposes exactly as though the conviction remained in place.

Similarly, a “deferred adjudication” whereby the record of the offense is expunged from the defendant’s record is nonetheless regarded as a conviction under U.S. immigration law.

If the foreign national has admitted to the crime, even if there was no conviction, he or she will be barred from receiving a visa (this can even occur during the medical exam preceding the consular interview if the unwitting applicant admits to prior substance abuse or some other crime).