Toothless' Saddle and Tail Fin

Due to the events of the first movie, Toothless was rendered unable to fly until Hiccup was able to replace the missing tail fin with a prosthetic one.

Appearance
Originally, the tail fin was a simple, brown-colored fan that was connected by a series of pulleys and cables, and controlled by a foot pedal located on Toothless' saddle. After it was destroyed during the battle with the Red Death, a more sturdier, red-colored version with the Hairy Hooligan insignia was created by Gobber.

Function
Before the movie, Toothless was able to fly by using his large wings for propulsion, broad maneuvers and lift, while his smaller tail fins allowed him to turn, and control altitude. However, when he lost his left tail fin, he became unbalanced and was unable to maintain stable fight.

After feeling guilty about rendering Toothless flightless and wanting to learn more about dragons, Hiccup constructed an artificial replacement for the missing tailfin. However, unlike the original one, Toothless could not deploy the new tail fin on his own, and requires Hiccup to work the controls. After several sessions of trial-and-error, Hiccup was able to work out the basic dynamics of dragon flight.

Basically, Toothless' tail fins work very much like the elevators and rudder of an airplane. Without Hiccup, Toothless is only capable of short hops or uncontrolled glides. For more complex maneuvers or true flight, he requires a rider, (normally Hiccup) to work the controls.

Modifications
The artificial tail fin has undergone several modifications. At one point, Hiccup decided to upgrade the tail fin mechanism to the point in which Toothless could fly on his own, (Gift of the Night Fury). However, this upgrade was short-lived as Toothless decided that he wanted to always be with Hiccup and smashed the new invention, thereby rendering himself dependent on his rider to fly.

Hiccup has been constantly modifying the tail fin to allow for faster speed, more maneuvability, and better response time. In the trailer for How to Train Your Dragon 2, it is shown that he has created a locking mechanism to keep the tail fin deployed, even when he was not on Toothless.