According to Air Canada, a fin number is a three-digit number on the tail of the aircraft. Aircraft — like cars — have some form of identification; in this case, the fin number is like a license plate for planes.
The fin number is used by airlines (such as Air Canada) as a form of unique registration. Air Canada in particular inscribes this registration number on the tail of the plane — also known as the fin (hence the name). Air Canada uses a three-digit number, but some other airlines use four digits.
The first number denotes the type of aircraft, while the second and third numbers announce the delivery date and model differences of that particular aircraft. Air Canada's website has a list of approximately 360 fin numbers.