Most 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 fingerprinted. 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.