Most airlines in the United States will not allow dogs to travel in the passenger portion (cabin) of the aircraft unless they will fit inside a carrier under the seat in front of them or on the passenger’s lap. Some exceptions are made for therapy dogs, sometimes also known as emotional or psychiatric support dogs, but these exceptions are completely dependent on the airline.
In general, most dogs are required to be placed in cages in the airline’s cargo area but always check with the airline prior to making any arrangements. Personally, I think it best not to transport your dog by air if at all possible.