Pernese Dragons
Pernese dragons are the largest land creatures on pern, fully sapient, and capable of communicating via telepathy. A dragon develops a bond with one person they meet upon hatching, whom they will remain with for the rest of their life and follow into death. Dragons were engineered from a much smaller native species by Kitti Ping, a brilliant geneticist and one of the first people to settle Pern. They were created as gigantic, flying flamethrowers to burn Thread spores out of the sky before it can reach the ground and begin consuming anything organic it comes in contact with. They can breathe fire only if they have ingested fire stone, a particular rock containing flammable gases, though the golden queens are incapable of processing it. Everything related to the origins of dragons, however, is lost history to the Pernese. They also don't know that fire stone doesn't cause infertility, it's just no one's been crazy enough to test that.
Thread falls on the planet on a daily basis when the planet it comes from, the Red Star, passes close to Pern in its orbit. But even without the threat of Thread, dragons are useful to have around for mundane tasks like transport. Not only do they fly, they are capable of traveling between. By hopping to a desolate and cold parallel dimension and back again, they seem to teleport.
While dragons seem very fearsome to most people, most dragons are incapable of harming another sapient being under most circumstances. (The chief exception is if two gold queens rise to mate at the same time, in which case they fight to the death.) Their telepathic senses make them very sensitive to the pain they cause. While they will go to extreme lengths in defense of their riders, the vast majority of dragons will feel horrible about it if they must.
Dragons mature fully at 24 months of age, though males might start participating in mating flights (probably unsuccessfully) at 18 months. (Canon has some extremely uncomfortable shit about dragonriders being compelled to have sex if their dragons mate. NOT IN THIS AU HOUSE.)
In appearance, they are generally smooth-lined with colorful, leathery hides. Pernese dragons have many-faceted eyes that change color depending on their mood (yes for real). They have almost giraffe-like "head knobs", which seem to have something to do with their psychic abilities, and the tips of their tails are forked.
Pernese dragons come in six colors, their color determining their size and general function in the Weyr. Colors are either all male or all female, and a simple mnemonic to remember is G colors for girls, B colors for boys.
Dragon spritework here is by Khiitan
From largest (and rarest) to smallest (and most common):

GOLD
Totaling in at less than 1% of the dragon population, gold dragons are also known as "queens". They are the largest dragons, and the largest land animals on Pern. Queens cannot chew firestone and produce flame, but they are also the only fertile females. Fortunately, queens are few and population is relatively under control. Gold dragons are afforded great respect and instinctive obedience by smaller colors, though this is not irresistable.
Gold dragons are a major determining factor in Weyr authority. A character that Impresses gold will, like it or not, be absorbed into the command structure of the Weyr for life. (Though extremely disruptive queenriders will be dealt with in some fashion or another. They are not above the law, and the senior queenrider will likely smack them down.) The rider of the Weyr's senior queen is called the Weyrkeeper, and serves as the Weyr's domestic head. Luckily, gold dragons seem to have a talent for picking out strong leaders. They seem to prefer less aggressive candidates than bronzes do, which suits them as gold riders and their dragons tend to serve as a pillar of stability for the Weyr. Weyrleaders come or go as the Weyr's senior gold dragon chooses mates, but the goldriders are forever.

BRONZE
Queens are the biggest dragons overall, but bronzes are the biggest males.They can even match their golden sisters in size, though it is very rare that a bronze reaches the maximum length. They are not as rare as gold dragons, but still very few in number. Bronzes are the only males big enough to have the stamina to keep up with a queen dragon for the full length of her mating flight, so they are the ones to sire clutches and always the color of a Weyrleader's dragon. This is helped by their natural authority over the smaller colors, like a queen's.
It's not as immediate with bronzes as it is with golds, but characters that Impress bronze will almost surely find themselves with a rank. Incompetence is not tolerated, but a young bronzerider has far less difficulty proving themselves as a capable leader than a rider of a smaller color purely on the merit of their dragon's shiny hide. It's unfair and old-minded thinking, perhaps, but that's the way the world turns. If a bronzerider's dragon catches the senior gold in a mating flight, the rider is the new Weyrleader, so it's beneficial for the rider to have had some experience with responsibility beforehand.
Bronzes are also good at picking out natural leaders, though they prefer a different breed than the golds seem to like. Fortune favors the bold, and so do bronzes. They typically seek daring candidates with the potential for courage under pressure, and those who are more willing to take risks. This not always the case, and even when it is it might not always be obvious to anyone (including the candidate themselves), but a bronze has a good sense for potential.

GREY
Greys are an all-female color and the only noncanon color included here. (Mostly, so people can play big female dragons without all the baggage that comes with gold.) They can rival bronze dragons and even the smallest golds in size, though usually are smaller. Like greens, greys will rise to mate several times a year but are incapable of laying eggs.
Greys are noted for greater independence than browns, typically. They're reputed to be a bit more aggressive, to have a little more "fighting spirit". This is not always the case, but often greys are noted for success working on the edges of a formation in roles requiring a bit more split-second decision making on their own.
Greyriders are less likely to be traditionally considered leaders, though a certain talent for rabble rousing is common. Greys tend to mark devil's advocates, "troublemakers", and otherwise unconventional souls. It's not rare to see a greyrider hold Wingsecond rank, but it will have been difficult for them to earn. Older riders might grumble about how this nontraditional nonsense wasn't tolerated back in their day.

BROWN
Brown dragons are the heavy workhorses of the Weyr. They make the best wingseconds' dragons and are generally steady in temperament. Occasionally, a brownrider might serve as a Wingleader if they show exceptional leadership ability. Browns almost never pursue queen dragons, as they are almost always too small and lack the long-distance flying ability to keep up with a Rising gold. They used to in days long past, but as dragons have gotten larger the golds have vastly outstripped them.
Typically brownriders are level-headed people, though there are exceptions to everything. The brownriders everyone imagines are responsible and steadfast, but perhaps lacking a certain ambition needed to catch the attention of a gold or bronze. Brownriders are also commonly chosen to serve as Weyrlingmaster, especially the patient ones.
Overall, the adage for the two big male colors goes like this: Bronzeriders are so crazy it just might work. Brownriders work so hard it just might be crazy.

BLUE
If it needs doing and isn't a matter of raw strength, a blue can probably do it. Even then, that's not to discount the muscle power behind an adult dragon. Blues are versatile workers, and while they can't fly great distances with heavy loads the way their larger cousins do they're still able movers of people and objects. Blues are also noted for their sensitivity, and along with greens they are preferred search and watch dragons. No blue dragon has ever so much as risen to pursue a gold, though they chase after greys readily and occasionally a big blue will manage to catch a small-end grey.
A bluerider is usually considered unremarkable by most weyrfolk. Unlike candidates that Impress one of the larger dragons, there are a lot of blues. Blues, like greens, are not very "picky" about their people.
Blues, like greens, are too small to have the stamina necessary to last for an entire threadfall. They will need to switch out for a reliefrider, or serve as someone else's.

GREEN
Greens are the smallest dragons, and therefore tend to be the best at maneuvering in general and especially in tight formations or small areas. All greens are female, but none of them are able to lay eggs. They are the most common color of dragon (roughly 50% of all dragons are green), and as such they do all sorts of jobs. Greens have stereotypically been considered duller than other colors, even ditzy, but it's really just that there are so many greens that the small percentage who aren't too bright make for a bigger number than other colors.
A green is a dragon that can do more with less. They require less food, less space, and less maintenance when young than the larger colors due to their size. They are even better search dragons than blues, unrivaled in their ability to sense children with the potential to Impress a dragon.
Green dragons have long been unsung heroes along with their blue brothers, essential to the efforts of a Weyr but often forgotten and left nameless in history as mere statistics.
Greens, like blues, are not especially picky about people and will cheerfully take anyone as a rider, occasionally being known to snipe a desirable candidate from under the nose of a gold or bronze. Also like blues, they are too small to have the stamina necessary to last for an entire threadfall. They will need to switch out for a reliefrider, or serve as someone else's.
White dragons are not accounted for in Weyr ranks, don't have a typical list of jobs, and aren't even expected to exist. They are rarer even than the golden queens, and are even smaller than the smallest green. Whites, when they appear, may be male or female. Neither sex is fertile, however, and mature whites express no interest in mating flights.
White dragons are a genetic fluke. The shell of a white dragonet's egg is unusually thick, making it extremely difficult for the baby to hatch out. Most die in the shell, onlookers never realizing that the apparent dud egg actually held a live dragon.
A white dragonet only survives if someone breaks the shell for it, a dangerous move under the eyes of a protective mother gold. Living white dragons usually survive because they manage to reach out to a person telepathically and call for help. Sometimes that person is a dragon candidate, sure, but a panicked dragonet will take anyone able to hear its cries.
All dragons are big and tough compared to most animals, but whites are fragile by dragon standards and are too small to do the strength and endurance work expected of dragonkind. There have been too few white dragons in Pern's history to draw many more generalizations about them than these.
It's a weird experience, being a whiterider. There is no older whiterider to mentor you. No one expects you, no one's sure what to do with you, and the jobs you were trained for (if you were even a candidate at all) just aren't something your dragon can always do. But they're still a dragon, you're still a dragonrider, and what are you going to do about it?

Each human figure is 6ft (~1.8m) tall, and each section is 10ft (~3m) square. (Also note that wingspans are probably a good deal larger than the artwork represents!)
Dragon lineart is by Sporelett
GOLD: 1-3
BRONZE: 1-4
GREY: 2-4
BROWN: 3-5
BLUE: 5-6
GREEN: 6-7