Toothless' Saddle and Tail Fin

Toothless' Saddle & Tail Fin were made for him due to the events of the first movie during a dragon attack on Berk. Toothless was rendered unable to fly on his own until Hiccup was able to replace the missing tail fin with a prosthetic one. To be able to fly and use the prosthetic tail fin, Hiccup must sit on Toothless, which resulted in Hiccup creating a saddle.

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.

The fin is pretty much remains the same after 5 years, except the symbol is a little different.

In Gift of the Night Fury, Toothless' old tail was replaced with a darker drown auto-tail that had no cables or whatsoever.

Toothless also had a greenish- yellow fin when he, Hiccup and Snotlout were stranded on Outcast Island, this being an 'emergency' tail Hiccup kept in his bag in case the usual red one was damaged and Hiccup cannot simply conduct repairs on Berk.

When Toothless' tail was damaged in Race to the Edge, Hiccup temporarily replaced it with the Dragon Hunters' flag. It was a slightly tattered beige flag with a red symbol. Upon seeing the symbol on his tail fin, most of the gang was in an uproar as they thought he had converted into a hunter. Later on in the series, he had a temporary blue tail fin with his face painted on it.

In the Film
Toothless' Tail is a special piece of equipment that Hiccup creates when Toothless loses his main left tail fin after being shot down. The design of the tail went through many variations until a working version was made.

The tail has a large leather saddle pad running down a part Toothless's back. This pad is clipped to two loops of curved metal that runs down and around Toothless's shoulders and front legs in order to keep it attached, secured with wide leather straps. Four lengths of cable made from ropes, two running down each side, allows Hiccup to adjust the prosthetic tail fin with control stirrups for his feet on either side of the saddle. Unfortunately, the tail was significantly damaged during the battle with the Red Death. When Hiccup wakes up back on Berk, both pieces of equipment have been remade by Gobber the Belch, with the left stirrup is replaced with a metal one that locks Hiccup's new metal foot perfectly into place, yet still allows him to control the fin. The new tail fin is red, and has a painting of a white skull on it.

Function of the Tail fin
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. The tail fin itself is a simple fan that is connected the the tip of the tail with a harness, and is controlled by a series of cables. The cables are powered by pulleys that lead to a stirrup on the left side of the saddle. By manipulating the stirrup, Hiccup can open and close the fan, depending on how fast or slow he needs Toothless to fly. With practice, any rider can gain enough experience to fly Toothless properly, though in Stoick's case, it was a rough ride for both him and Toothless.

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 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 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. However, this lock does not allow Toothless make turns or change directions easily. Toothless is still able to dive and glide with the locked tail, as seen in the second film.

History
Hiccup first made the tail fin after he realized Toothless was injuried. The first versions were unstable and hard to control, they were "flopping" around and wouldn't stay flat which forced Toothes to constantly counter it. With every new version it was gradually improved. While final version of the tail was a perfect fit, he still needed help controling it. Hiccup also made a saddle for Toothless so he could help the dragon fly again. During the battle with the Red Death, it set the tail fin on fire, slowly burning it up. Gobber later made a new tail and fixed the saddle afterwards.

Gift of the Night Fury
In the exclusive short Gift of the Night Fury, Hiccup makes a new prosthetic mechanical auto-tail that doesn't require Hiccup to control, a comment made by Tuffnut prompting Hiccup to reflect that it is unfair that Toothless is totally dependent on him for flight.

This new tail consists of a lever that attaches to the leading edge of Toothless' right tail fin so that he can control the left concurrently.

When his right tail fin is extended, so is the lever which is attached to a rotating wheel that rotates approximately 180 degrees. Another metal rod is attached from the rotating wheel to an 'egg-shaped' piece. A gear shaped like a fraction of a circle is attached by an axle on top of the 'egg-shaped' piece so that the rotation can turn the gear along the five cogs. The gear is meshed perpendicularly with a full-circle gear that will rotate and extend the tail fin.

This tail fin seems to not allow special maunevers like a barrel-roll while flying forward which is due to the fact that the tail to work, it depends on the other tail. Most special maunevers require both tails be in different positions to work and thus can't be peformed by this tail. Toothless was able to use this tail fin to fly off and find Hiccup's helmet, while he had lost the day before.

Near the end of the special however, Toothless insists on using the saddle and stirrup system and when Hiccup doesn't understand why, he destroys the automatic tail by whipping it several times to the ground and flicking it away. He then nudges the saddle towards Hiccup, showing that Toothless would rather fly with his friend than have the freedom to fly on his own.

Another reason that Toothless may have thrown away the new tail may have been he didn't feel comfortable with the new tail design.

DreamWorks Dragons: The Series
The saddle and tail continue to be used in Riders of Berk, although it uses the same mechanics as the auto tail, but still requires Hiccup's help.

Hiccup briefly returned to a variation of the original stirrup setting when he was showing Stoick the advantages of owning a dragon by lending him Toothless, allowing Stoick to control Toothless' tail.

Toothless's tail has gone through modifications in the series as well. During the Thawfest Games, Hiccup made a new tail that was "as thin as paper" to help in their race against Snotlout, but it was still the same design as the red one.

Toothless lost his tail in a storm and he, Hiccup and Snotlout were stranded on Outcast Island. Luckily, Hiccup brought a spare yellow tail with him, but the connecting rod was broken, requiring them to sneak into the Outcasts' village so that Hiccup could acquire a replacement.

Following his attempts to 'train' Stoick, it would appear that Hiccup has modified Toothless's stirrup so that it can be used by other riders as well as by himself, as Fishlegs and Tuffnut and Ruffnut have each been able to ride on Toothless after their dragons accidentally flew off with Hiccup without changing the usual saddle setting, even if their control was limited to just keeping Toothless up in the air.

How To Train Your Dragon 2
The saddle seen in this movie is of a notable different design. It is longer so Hiccup can sit and lay on Toothless in different positions while flying, and also features a function which allows Hiccup to lock the artificial fin's position to the natural one, in a similar way to the auto-tail. This was used when Hiccup used his skydiving suit, letting Toothless fly alone and others to ride him, even if they are not familiar with this saddle and normally wouldn't be able to use it; Drago Bludvist was even able to use the saddle when Toothless was briefly under the control of Drago's Bewilderbeast, although he was never shown attempting any particularly elaborate maneouvres.

This mode however, still has problems of the previous versions, limiting Toothless' manoeuvrability as most of the advanced maneuvers require the fins to be in different or opposite positions, instead of mirrored, which poses problems when he is forced to quickly catch Hiccup while he is skydiving. This problem was later reduced, but not totally eliminated, when Valka shows him Toothless' hidden fins that gave him even more control in flight.

The saddle also features more pockets and different handles, to let Hiccup hold it in different positions.

The stirrup is also heavily modified allowing for more control using Hiccup's prosthetic, which also was modified with two switchable modes, one for walking, and one for flying.

The design of the skull is also different, having a lower jaw and matching up with the symbol on Hiccup's right shoulder pad.

When Valka sees the tail fin, she first thinks the loss of the original fin was the work of Drago. Hiccup however awkwardly explains he was the one who shot Toothless down. He also adds how they were even after Hiccup lost his leg.

Trivia

 * In the movie, the skull is on both sides of Toothless' tail. In the rest of the franchise, it is just on the front.