What are the fins near the tail of a helicopter called? My young sons are very enthusiastic about helicopters and I am repeatedly asked to name parts of their model aircraft. I would like to do so correctly, but my research has not always given definitive answers. I hope you can help.

Fixed to the tail boom and close to the tail rotor, some helicopters have a pair of horizontal fins. My research suggests these could be referred to as elevators, horizontal stabilisers, or horizontal fins. Is there one correct term, or does it depend on how they are used? Why do some helicopters not have them?

Most (all?) helicopters have a vertical fin connected to the tail boom. I have seen this labelled as a rudder or vertical fin. Does this depend on its function (moveable or fixed?) or can you suggest a better name?
You can call them horizontal and vertical stabilizers, since that is exactly what they do, in forward cruising flight.  The vertical stabilizer is there because the tail rotor itself doesn't provide as strong a weathervane effect as a fixed surface. So it stops the nose from oscillating left and right in forward flight same as the stabilizer in an airplane.  

The horizontal stabilizer is to provide some down force to keep the body angle a little more level in cruise flight.  The rotor disc is flying along doing its thing with this big weight under it, which wants to swing back and pitch the nose down from air drag and the forward movement of the suspension point (more or less like running along with a tennis ball suspended on a string - it swings back and stays at an angle until you stop running).  The horizontal stabilizer's down force counteracts some of this and helps dampen pitch oscillations and keep the nose at a more desirable (higher) attitude.