The following definitions are paraphrased from Title 47 - U.S. Code of Federal Regulations, Part 97 (Amateur Radio Service) and apply to all U.S. licensed amateur radio operators. Please refer to the actual text if you have any questions. Non-U.S. participants must comply with the rules of their appropriate governing organization for amateur radio.
1. Control operator. An amateur operator designated by the licensee of a station to be responsible for the transmissions from that station to assure compliance with the FCC Rules. (Reference: §97.3.(a)(12))
2. Third party communications. (Reference: §97.115 (a) and (b))
a. An amateur station may transmit messages for a third party to any station within the jurisdiction of the United States or any station within the jurisdiction of any foreign government whose administration has made arrangements with the United States to allow amateur stations to be used for transmitting international communications on behalf of third parties.
b. No station shall transmit messages for a third party to any station within the jurisdiction of any foreign government whose administration has not made such an arrangement. This prohibition does not apply to a message for any third party who is eligible to be a control operator of the station.
Technician class operators without HF are limited to making contact only with those countries which share a third party traffic agreement with the U.S. Technician class operators with HF privileges can make unrestricted HF contacts on 10M.
c. The third party may participate in stating the message where the control operator is present at the control point and is continuously monitoring and supervising the third party's participation.
Note: The third party may not be a prior amateur service licensee whose license was revoked; suspended for less than the balance of the license term and the suspension is still in effect; suspended for the balance of the license term and re-licensing has not taken place; or surrendered for cancellation following notice of revocation, suspension or monetary forfeiture proceedings. The third party may not be the subject of a cease and desist order which relates to amateur service operation and which is still in effect.
d. At the end of an exchange of international third party communications, the station must also transmit in the station identification procedure the call sign of the station with which a third party message was exchanged.
3. The list of countries with which the United States has a third party agreement can be found here.