Typically, you will need local and federal clearance to adopt a child. This will vary from country to country. In the U.S., for example, you will be subject to a background check by both your state and the FBI. If you have an arrest on your record, you will need to write a letter explaining it.
About 135,000 children are adopted in the United States each year. Of non-stepparent adoptions, about 59% are from the child welfare (or foster) system, 26% are from other countries, and 15% are voluntarily relinquished American babies.
Most foreign-born adopted children acquire U.S. citizenship as soon as they enter the United States, thanks to the Child Citizenship Act of 2000 [PDF] (U.S. Citizenship and Immigration Services). Since the passage of the law, they do not need to formally apply but become citizens automatically.