When a child leaves the country with only one parent, carrying documentation that proves the other parent is aware and approves of the travel is strongly recommended by U.S. Customs and Border Protection. The statistics surrounding child abductions and trafficking have made government agencies more aware of minors crossing borders. You may not be asked to show any kind of documentation, but it's a good idea to have it, especially if you share custody with the parent who is not traveling.
Notarized Acknowledgment Letter
The parent not traveling with the child should write a note that acknowledges international travel is occurring and provides permission. The Department of Homeland Security recommends the letter state the name of the traveling parent and the name of the child. You should include the exact dates of travel and the acknowledgement letter should be signed. It might read something like "I acknowledge that my ex-wife, Mary Smith, is traveling to Paris with our 4-year-old son, Jacob Williams, from May 10 to 24, 2014. She has my permission to travel with my son." You also need to sign and date the acknowledgement. U.S. Customs and Border Protection recommends you have the form notarized.
A single parent who has sole custody and is traveling internationally with his children should bring a copy of the custody documents. If you are not in touch with the other parent or don't know where the other parent is, a court document that lists you as sole custodian can replace an acknowledgment letter from the other parent. A divorce agreement that lists the custody arrangement also may work if you're required to prove you are entitled to travel out of the country with your child.
A birth certificate may seem unnecessary since your child is required to have a passport, but it can help you establish yourself as the child's parent in case there are questions. For example, if your last name is different than your child's last name, the birth certificate lists you as one of the parents. The birth certificate can verify the person who wrote and signed the acknowledgement letter is the other parent.
Requirements of Destination Country
The country you are traveling to may have specific requirements when it comes to minors entering with only one parent. Check the U.S. State Department's page entitled "Learn About Your Destination" for country-specific information on the documentation and forms you may need to have with you.