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Berk Dragons in general:

Although not always well behaved, the Hooligan Tribe's dragons have made themselves at home in Berk and live with them. Naturally this was not a smooth change, but they have integrated with their human allies.

All Animals Are Domesticated: Averted. There were many headaches in Berk after they moved in, forming many of the plots of Riders of Berk, but eventually they settled down and integrated into society.
Dragon Rider: The dragon half of the equation.
Heel-Face Turn: Were originally Berk's enemies, until Hiccup and Toothless destroyed the Red Death.

Toothless alone:

Both Universes -

Adaptational Badass: In the books he's a Basic Brown/Garden Dragon, one of the smallest, most common dragon species (the template for the Terrible Terrors in the movies) while in the films he's a Night Fury, a Lightning Bruiser phantom dragon nicknamed "the unholy offspring of Lightning and Death." Book 11 reveals that the book series Toothless is an adolescent Seadragonus Giganticus Maximus, the same species that the Big Bad of the first film was based off of. It is further revealed that Toothless will grow up to be the size of a mountain, dwarfing even the Alpha dragons of the films.
Defrosting Ice Dragon: To say he was hostile to Hiccup at first is an understatement. In the books, he is selfish and mean to Hiccup, eating his food and not listening to him, however in the climax he flies up a big dragonus giganticus maximus's nose, almost killing himself, to save his trainer.
Determinator: Nothing stops him from protecting Hiccup; not a cliff, not a chain and especially not other dragons.
Heterosexual Life-Partners: An interspecies example. With Hiccup, a Viking human.
Tsundere: Toothless likes to switch among affectionate, aloof, and hostile at the drop of a hat.

Film & TV Universe -

Adorkable: "The Offspring of Lightning and Death" is also one of the biggest dorks.
All Animals Are Dogs: Downplayed; Toothless does have a few canine traits, like his habit of enthusiastically licking Hiccup in the second movie, but much of his behaviour is actually based on a cat's and his design was modeled after a black panther.
Arch-Enemy: With the Whispering Death in "What Flies Beneath".
Artificial Limbs: Toothless has a very complicated artificial tail fin made for him by Hiccup.
Asskicking Equals Authority: Becomes Alpha Dragon by challenging and defeating the reigning Alpha. Said reigning Alpha is a massive Bewilderbeast that is hundreds of times bigger than Toothless and has mind control powers. Toothless defeats him in a head on fight!!
His first act as Alpha? Command the swarm of dragons to blast the Bewilderbeast into submission!
Badass: The premier example of Badass in this series, aside from Stoick.
Badass Adorable: A fierce creature that likes having its belly rubbed.
Badass In Charge: Becomes the Alpha Dragon at the end of the second movie.
Badass in Distress: He gets mind-controlled by the Dark Alpha during the beginning of the attack on Berk and Drago seizes him as a mount. Luckily Hiccup manages to get through to him.
Berserk Button:
Don't hurt Hiccup if you know what's good for you—he made sure a Monstrous Nightmare got the message.
He really hates the Whispering Death. Just its presence makes Toothless fly into such a rage to the point he even disobeys Hiccup.
In the second film, Hiccup's put through the ringer enough to enrage Toothless to the point that he takes on the Bewilderbeast from the ground. This thing is the size of the Red Death, and Toothless specifically takes it on without Hiccup, so he can't fly or outwit his enemy. No tricks, no traps, just pure fury and willpower enough to free the dragons from the Bewilderbeast's control. If it's not clear yet, do. Not screw. With Hiccup.
Beware the Nice Ones: When he's not in battle, he's like a cuddly pet.
Big Good: Becomes the Alpha of the dragons by the end of the sequel, with Hiccup as the new Chief of Berk.
Bioluminescence Is Cool: His back begins glowing in the climax of How to Train Your Dragon 2.
Breath Weapon: Night Fury's have a unique one that launches small precise blue fireblasts that explode when they hit something, as opposed to a stream of fire, which proves far more destructive than the more traditional flames. He can also spit narrow jets of flame, which he uses to make a nest early in the first film.
Cute Bruiser: A cuddly reptile that is also the most feared dragon.
Dark Is Not Evil: Movie only, where he is a Night Fury, a dark coloured species with a semi-demonic look.
Deuteragonist
The Dreaded: Implied to be this to (most) vikings by The Dragon Manual.
Empathy Pet/Uncatty Resemblance: To Hiccup, being (comparatively) small, smart, snarky, and deceptively strong. According to Valka, they're even in roughly the same age range!
Expy: In terms of face and limb structure, he looks a lot like Stitch.
In the finale of the second movie with his spine glowing blue and becoming the undisputed 'King' of dragons, you'd be forgiven if you started calling him Godzilla.
Extreme Melee Revenge/Roaring Rampage of Revenge: Cue Toothless taking on and beating the crap out of said Bewilderbeast, then usurping the Alpha role from it and using it's newfound power to deliver a No-Holds-Barred Beatdown upon the giant dragon and it's trainer.
Fluffy the Terrible/Deathbringer the Adorable: His personal name is the former but his species name is the latter.
Game Face: Toothless extends his teeth when eating or being aggressive.
Genius Bruiser: Night Furies are the cleverest dragon species.
Guile Hero: Toothless must've been taking lessons on this from Hiccup, as in the season one finale, he stops eating and thereby forces his captors to remove his muzzle to try to force feed him. He immediately hits them with a plasma burst and tears off his remaining restraints, then swats the guards with his tail.
Handicapped Badass: In the films, where one of his tail fins is lopped off by Hiccup early on.
Hellish Pupils: Deliberately invoked: when he becomes angry his pupils narrow into vertical slits, but when he's calm and happy they're round.
Informed Ability:
In the movie, Night Furies are said to be the fastest and never miss their mark. In the series, Toothless has missed a few times and Stormfly is said to be getting closer to his speed. Toothless does have a handicap, though, and by and large every target in the film was stationary (relatively or otherwise).
Likewise, both the tie in ads and Defenders of Berk states that Night Furies only have six shots. Toothless regularly unleashes far more in a fight.
Last of His Kind: Implied to be the last remaining Night Fury.
Let's Get Dangerous: When they encounter Eret's dragon-hunting crew, Toothless goes into screeching, plasma-shooting blur mode... which is significantly less effective in broad daylight.
Lightning Bruiser: He is a Night Fury after all, and is no exception to the species.
Monster Lord: By the end of HTTYD 2, Toothless is the new dragon Alpha.
Moody Mount: Sometimes he'll decide, on a whim, 'I want to barrel roll' and Hiccup will have to hold on.
Non Indicative Name: His teeth are retractable, not missing.
Panthera Awesome: Has several characteristics of a panther and even moves around like a panther.
Pint-Sized Powerhouse: Toothless is one of the smallest dragons that can be ridden, and he doesn't measure up to many other dragons in terms of generating actual fire. But his unique form of fire breath lets him generate small, precise shots with extreme explosive force, more like an artillery strike than a blast of fire. This lets Toothless damage other dragons, which are immune to fire, with his shots, up to and including the Red Death and the Bewilderbeast, two dragons that dwarfed Toothless like an elephant dwarfs a rat. Despite his size, he is much stronger than he looks, tossing off a Monstrous Nightmare and also pinning Stoick.
Power Glows: See Bioluminescence Is Cool.
Red Is Heroic: His normal tailfin is with Berk's symbol also painted white on it.
Revenge Before Reason: Wouldn't even let Hiccup get on him to let him fly in "What Flies Beneath" because of his grudge with a Whispering Death, even though it meant he couldn't fly and was left at a serious disadvantage.
Ridiculously Cute Critter: Toothless is essentially a giant, winged axolotl, complete with "gills" reimagined as an expressive fleshy crest on his neck.
Super Speed: One of the fastest species of dragons.
Shipper on Deck: The look on his face after Astrid pecked Hiccup's check in the movie.
Silent Snarker: In the movie. He is the most expressive dragon, primarily due to his size-changing pupils. He takes after a certain someone.
Throat Light: His Super Mode involves his back glowing, as shown in the climax of How to Train Your Dragon 2.
Troll: When camping with Hiccup, Toothless snatched a fish from a Terrible Terror, deliberately provoked it into trying to attack him, then shot a fireball down its throat, smiling smugly the whole time.
Unstoppable Rage: We see a few glimpses of how terrifying he can be when his trainer/friend Hiccup is threatened. First one was when Hiccup was tackled to the ground by Astrid when she sees Toothless nearby. Toothless, taking it as Astrid attacking Hiccup, goes absolutely mental and charges at her, Hiccup snaps him out of it but even as he was introducing Astrid to him Toothless was giving her an absolutely terrifying Death Glare as well as a snarl to remind her that he still wasn't finished with her. Later when Hiccup's life is in danger in the fighting ring, Toothless hears his cries, proceeds to speed through the forest, jump through into the ring and deliver a Curb-Stomp Battle on the offending dragon. Also in the second movie when Drago orders the Bewilderbeast to end Hiccup, Toothless dives to take the blast, then blows the ice prison up, declaring to all that his Berserk Button is well and truly pressed. He then channels a Godzilla impersonation that is far more accurate than a twenty-foot-long animal has any right to pull off.