Many States require by law that criminal record and child abuse record clearances be conducted on all adoptive and foster parent applicants. This usually involves filling out a form with your name (in a woman's case, it would include her maiden name and former married names, if applicable), date of birth, and Social Security number; possibly getting the form notarized; and sending it to the State child welfare and police agencies for clearance. In some States it might involve being finger printed. The authorities will check to see if you have a child abuse or criminal charge on file.
Misdemeanors committed long ago for which there is a believable explanation (for example, "I was young and foolish and did what the guys expected me to...") usually are not held against you. A felony conviction, or any charge involving children or illegal substances, would most likely not be tolerated.
Source: National Adoption Information Clearinghouse