Toothless' Saddle and Tail Fin

Toothless' Saddle and Tail Fin is an artificial prosthetic tailfin that Hiccup designed and built for Toothless. It consists of a leather and metal tailfin controlled by a series of cables, pulleys, and gears connected to a set of pedals which Hiccup controls with his feet while sitting on a saddle.

Origin
Toothless lost his left tail fin after Hiccup shot him down during a dragon attack on Berk at the beginning of the first movie and was hence rendered unable to control his flight, leaving him unable to fly properly. After Hiccup freed Toothless and realized his tailfin was missing, he decided to make a prosthetic tail for the dragon.

The tail's first iteration consisted only of a strap-on prosthetic tailfin; after his first (unsuccessful) test with Toothless, Hiccup discovered that the tailfin's position needed to be varied in flight, which he attempted at first by attaching the fin to a rope and manipulating it by hand before eventually designing a gear system allowing him to control the tailfin with his left foot while sitting on a saddle.

The original fin was destroyed in the battle with the Red Death; Gobber built a second fin with a modified pedal for Hiccup's new prosthetic (him having lost part of his left leg in the attack). This fin is rebuilt and modified many times by Hiccup over the years, culminating in the design shown in How to Train Your Dragon 2.

Design
The tail fin itself is, essentially, a fan-shaped like a Night Fury's tailfin, with leather held together by metal rods. Its original iteration consisted of merely the tail fin portion; eventually, the whole thing evolved to include a series of gears, pulleys, and cables controlled by a foot pedal mounted on a saddle which Hiccup sat on and controlled the pedal from. The original tail was colored brown, likely as it was made of leather - after it was destroyed during the battle with the Red Death, Gobber built a new version with a modified pedal designed for Hiccup's prosthetic, with a red-painted fin with a white skull symbol as shown in How to Train Your Dragon.

In Gift of the Night Fury, Toothless' old tail was replaced with a darker brown auto-tail that had no cables or whatsoever and gave him the ability to fly on his own.

Toothless also has a greenish-yellow 'emergency' fin. It is seen (and used) in "Defiant One" when Hiccup and Snotlout were stranded on Outcast Island, Hiccup having kept it in his bag in case the usual one was damaged and Hiccup couldn't conduct repairs on Berk.

When Toothless' tail was damaged in "A Time to Skrill", 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, many of the Dragon Riders were in an uproar as they thought he had become a hunter. Later on in the series, he had a temporary blue tail fin with his face painted on it.

In How To Train Your Dragon 2, the tail fin is shown to have been upgraded significantly. Notably, it now has a much more complex control system, featuring two pedals - one on its which controls the fin's vertical angle/movement, and a 'slider' pedal on the left which controls the fin's outward/horizontal extension. It also now features a locking mechanism in the form of a lever allowing for the tailfin to be locked in an open position to allow Toothless to glide while Hiccup uses his wingsuit.

Brown
The original tail. It is colored brown, likely as it is made out of leather. Several variations of this tail were produced until Hiccup learned the basics of dragon flight and found a working tail design. The tail's first iteration consisted only of a strap-on prosthetic tailfin; after his first (unsuccessful) test with Toothless, Hiccup discovered that the tailfin's position needed to be varied in flight, which he eventually did by pulling a rope by hand and eventually designing a gear system allowing him to control the tailfin with his left foot. However, it was still flawed - even the latest iteration used (before it was destroyed) was too 'floppy' in the air, constantly flipping up and down due to the wind, meaning Toothless had to constantly counter what the prosthetic was doing. This tail was destroyed in the battle with the Red Death.

Red I
Built by Gobber while Hiccup was recovering after losing his leg, this tail was much stronger and sturdier then its predecessor. It had a modified stirrup designed for Hiccup's original prosthetic and is painted red with a skull symbol on it. It has been rebuilt many times.

Automatic (Brown)
On the first Snoggletog with dragons, Hiccup built Toothless a new tail that would allow him to fly on his own by using a complex gear system to mirror the right tail fin. This allows Toothless to fly on his own but means he only has basic control. After retrieving Hiccup's helmet, Toothless destroyed the tail preferring to fly with Hiccup then on his own. It is possible that, if the fin had survived, Hiccup would have added a manual stirrup to allow him to bypass the auto gear system to allow for more complex maneuvers, especially when in battle.

Red II
Made for races, this tail is as thin as paper and stronger than any of its predecessors. It means Toothless can cut and turn almost as if he had his own tail fin. This upgraded version first appeared in How to Train Your Dragon 2.

Yellow
This tail fin acted as a spare tail during the events in Dragons: Riders of Berk episode "Defiant One". Hiccup cobbled together this tail together when he and Snotlout crashed on Outcast Island. A connecting rod for Toothless' primary tail was damaged in the crash-landing, and things were further complicated when Snotlout broke a replacement connecting rod.

Yellow II
Made of a thin sheet of Gronkle Iron, this tail is not well suited for long flights but perfect for gliding stealthily under a boat's sightline for a sneak attack.

Blue
Flimsy and sensitive, this tail allows for tight turns in small spaces but not high altitude.

Green
Made of Gronkle Iron, this tail is used in hot places such as volcanoes. Hiccup created this tail fin, along with others, during the events of Dragons: Race to the Edge. This green tail was destroyed trying to get an Eruptodon egg to its sacred nesting ground inside the active volcano on Caldera Cay, in the episode, "Out of the Frying Pan".

Red III
The latest iteration of Toothless's tail, this tail is an evolution of all the past tails. It features a much more complex gear system - unlike the older tails, this tail has two 'pedals', one on its right which controls the vertical angle of the tail, allowing for greater control to do things such as barrel rolls and tighter turns, and a 'sliding' pedal on its left controlled by Hiccup's prosthetic controlling the outward (horizontal) extension of the tail. This duel controlled tail allows Hiccup to more precisely control the tail to allow for more precise movements. It also has a locking mechanism allowing for Toothless to glide alongside Hiccup as he uses his flight suit but this greatly limits Toothless' movements making it very difficult for him to rescue Hiccup from dangerous situations but, with the discovery of his split back fins, it is easier for Toothless to maneuver and save Hiccup.

Trivia

 * In the film, the skull is on both sides of Toothless' tail. In the rest of the franchise, it is just on the front.
 * The tail fin from the second film appears in the game, Dragons: Rise of Berk as a collectible in completing collections.
 * The symbol on Toothless' tail fin in How to Train Your Dragon 2 is similar to the symbol on Steve Plachuk's shirt from Trollhunters: Tales of Arcadia.
 * In the 'what-if' scenario imagined by Astrid in "Darkest Night" where she wondered what would have happened if Hiccup had never shot Toothless down, the saddle and fin were devised by Gobber while Hiccup was unconscious after Toothless saved his life following a confrontation with Toothless' Nemesis.
 * In Gift of the Night Fury, Toothless prefers to use his old tail fin rather than the new one Hiccup made him enable him to fly on his own.