I’ll list them by size.
Balerion - Largest of the dragons, he died a few years before the show House of Dragons started. He was originally flown by Aegon the First during the initial conquest of Westeros. His last rider was the current king, Viserys.
Vhaghar - A giant ancient dragon almost as big as Balerion at his peak and maybe older than Balerion when he died. In the past she was ridden by one of Aegon the First’s SisterWives. In the show she will be ridden by Laena, the 12 yo girl being shopped to be the second Queen of Viserys. But Laena dies giving birth to a dead son of Daemon. So eventually Aemond (second son of Viserys and Alicent Hightower) takes Vhaghar as his Mount. First the two kill Rhaenyra’s second son Lucerys and his dragon Arrax during a diplomatic mission which essentially kicks off the kinslaying for real. Next the two team up with Aemond’s older brother King Aegon the Second and his dragon Sunfyre to fight Rhaenys, the Queen Who Never Was, and her dragon Meleys. They successfully kill Rhaenys and Meleys but King Aegon the Second is horribly burned and forced to give up the throne to Aemond and Sunfyre is severely injured. Aemond and Vhaghar are eventually killed by Daemon and his dragon Caraxes (either all four die or only Daemon survives depending on which narrator you believe).
Meraxes - Ridden by one of Aegon the First’s SisterWives, she was killed by a scorpion bolt shot by Dornish soldiers during their failed invasion of Dorne as part of the Conquest of Westeros. So she is dead long before the new tv series starts.
The Cannibal - A wild dragon who is never tamed and survived the Dance of Dragons. It disappeared from Westeros and no one knows what happens to it.
Vermithor - Ridden by King Jaehaerys (the father of King Viserys) until the king’s death, Vermithor remains riderless until Rhaenyra‘s faction, the Blacks, decide they need additional dragons. Several men were killed trying but eventually an unrecognized Targ bastard Hugh Hammer managed to win him as a Mount. After a few battles solely against human foes, Hugh switched allegiances along with another Targ bastard dragonrider to the Greens in the midst of a battle and the Black forces were devastated. Later Hugh was caught separated from his dragon and killed on the ground while the riderless Vermithor was attacked in the air by Seasmoke and a legitimized bastard of Corlys (claimed by the gay Laenar) as well as Tessarion and Daeron. All three dragons and both riders were killed in the battle.
Sheepstealer - A wild dragon claimed by a female Targ bastard named Nettles. She was accused of being a traitor like the other bastards by Rhaenyra so she fled and hid away with her dragon (and possibly her new lover Daemon).
Silverwing - Riderless at the start of the show, a Targ bastard named Ulf White becomes her rider. After some battles against just men, he turned traitor and left the Greens. Ulf is poisoned and Silverwing becomes riderless through the end of the House of Dragons and is one of only four dragons to survive although she goes wild.
Dreamfyre - Ridden by Helaena, the second daughter of Viserys and Alicent and sisterwife to Aegon the Second. Never fights in the civil war until men storm the dragon roost and Dreamfyre is killed along with three other dragons by humans.
Caraxes - Ridden by Daemon, participates in lots of momentous battles and events but the only dragon fight is against Vhaghar and King Aegon the Second where either all perish or else Daemon is the sole survivor.
Meleys - Ridden by Rhaenys, the Queen Who Never Was, they are both killed by the combined efforts of Vhaghar, Sunfyre and their riders.
Syrax - Rhaenyra’s mount, she takes part in battles against men but never against dragons. When the population of Kings Landing rises up against the conquering Blacks, Rhaenyra is too sick to fly following a birth and her youngest son Joffrey tries to fly on Syrax to attack the rioters and save the other dragons but falls off to his death. Syrax dies riderless, killed by rioters.
Quicksilver - Died before House of the Dragons in a previous Targ Civil War by Balerion.
Gray Ghost - Wild, riderless dragon eaten by a temporarily riderless Sunfyre during House of the Dragons.
Sunfyre - Ridden by Aegon the Second, they participate in a battle with Vhaghar and Aemond where they kill Rhaenys and Meleys but Sunfyre and rider are severely injured. Sunfyre is first guarded by Green soldiers on the battlefield but eventually is healed enough to fly to King’s Landing. When the Blacks manage to take King’s Landing, Sunfyre flies off by his own. When the wild unclaimed Dragon Gray Ghost was seen being killed by another riderless dragon, most people suspect it was the wild dragon The Cannibal but Aegon the Second believes otherwise and is reunited with Sunfyre which helps heal him. During a later battle, Aegon the Second and Sunfyre fight Baela (daughter of Laena and Daemon) and the tiny dragon Moondancer. Although they win, Moondancer manages to severely injure Sunfyre again. Sunfyre eventually dies of these infected wounds months later but not before Aegon uses him to execute his half-sister and main rival Rhaenyra.
Tessarion - Ridden by Daeron, third son of Viserys and Alicent. They are killed along with all other dragons and the rider Addam in a fight with the riderless Vermithor against Seasmoke and Addam.
Seasmoke - Ridden by Addam Velaryon a bastard of Corlys legitimized by the gay Laenor, they are killed in a fight with the Riderless Vermithor as well as Tessarion and Daeron but they manage to kill their opponents as well.
Vermax - Ridden by Jacaerys, firstborn son of Rhaenyra and allegedly the gay Laenor, they are killed attacking a Green fleet.
Arrax - Ridden by Lucerys secondborn son of Rhaenyra and allegedly the gay Laenor, they are killed on a diplomatic mission by Aemond and Vhaghar.
Tyraxes - Ridden by Joffrey thirdborn son of Rhaenyra and allegedly the gay Laenor, Tyraxes is killed riderless by rioters in King’s Landing.
Morning - Ridden by Rhaena, daughter of Daemon and Laena, they survive House of Dragons without fighting.
Stormcloud - Ridden by a very young Aegon the Third, son of Rhaenyra and Daemon. Stormcloud is killed by arrows escaping with Aegon III which is their first ride. Aegon III survives.
Shrykos - A young riderless dragon who is killed by rioters along with three others in King’s Landing.
Morghul - Same as Shrykos. A young riderless dragon who is killed by rioters along with three others in King’s Landing.
Moondancer - The smallest dragon actually ridden. Her rider is a 13yo Baela, daughter of Daemon and Laena. Moondancer is killed in a battle against Aegon the Second and Sunfyre but her rider Baela survives and Sunfyre is mortally wounded as well while Aegon the Second is severely injured.
Drogon, Rhaegal, & Viserion - The three dragons of Daenerys from the Song of Ice and Fire. They are depicted in the graphic as the size they are in the last finished book. However, using a size comparison on the tv show, Drogon ended up being almost exactly half the size of Balerion which is still a huge dragon. Moreover, Balerion continued to grow throughout his life just like sharks and alligators and while the growth slowed, he grew over his entire 200+ year lifespan.
The Last Dragon - Dies young from being born twisted and mutated after House of the Dragons.