Meleys was one of the dragons that belonged to House Targaryen, which fought during the great civil war known as the Dance of the Dragons, during which it was the mount of Rhaenyra Targaryen's former mother-in-law Rhaenys Targaryen.
During the Dance, Meleys was the mount of Rhaenys Targaryen, known as "The Queen Who Never Was".
In the opening moves of the war, Rhaenyra's rival Aegon II Targaryen and his new Hand of the King, Criston Cole of the Kingsguard, led their army to attack Rook's Rest, a castle on the mainland north of Rhaenyra's stronghold on Dragonstone which had declared for her.
Besieged, the castle's lord sent messenger-ravens desperately begging for Rhaenyra to send aid. When help did arrive it was not an army, however, but Rhaenys riding her large red dragon Meleys, which proceeded to burn many of the soldiers besieging Rook's Rest.
It was all a trap, however, as two enemy dragons suddenly confronted Meleys: Aegon II riding Sunfyre, and Vhagar - the largest and greatest living dragon - ridden by Aegon II's younger brother Aemond Targaryen.
Meleys had no hope of winning against both of them, but to her credit Rhaenys did not flee, instead turning Meleys to meet them head-on. In the massive mid-air battle that followed dragon fought dragon, and their flames burned so brightly it was as if a second sun blossomed in the sky.
When the smoke of their confrontation finally cleared, only Vhagar and Aemond were not seriously wounded. Meleys was dead, torn to shreds, and her rider Rhaenys burned to ash. Yet Meleys did not die without exacting a price: she had maimed Sunfyre, almost completely tearing one of his wings from his body. Aegon II himself was gravely wounded, crushed under his dragon and severely burned over half his body. He survived, but had to be sedated with Milk of the poppy, and remained hovering out of consciousness for months. This left Aemond to command Aegon II's faction while his brother convalesced - another serious blow, as it turned out, for Aemond was hot-head and poor strategist, easily fooled into leaving King's Landing poorly defended.
In the books
In the prequel novellas, Meleys was known as "the Red Queen", which seems to indicate that she was the mother of several of the other dragons (much as Rhaenys was grandmother to the youngest generation of dragon-riders). Her exact age and lineage has not yet been established.
Meleys does appear to have been one of the oldest and largest dragons after Vhagar herself, and was probably one of the second generation dragons hatched to the original three from the Conquest-generation (Balerion, Meraxes, and Vhagar). Vhagar was 181 years old during the Dance, and by that point as said to have grown as large as Balerion had been during the War of Conquest. Vhagar is also described as being five times the size of the young dragon Arrax, who was only a decade or so old and only just big enough to ride to war. It was widely believed that no dragon could defeat Vhagar in a one-on-one fight - even Daemon doubted if Caraxes could face her and live (and indeed, only later managed to defeat Vhagar in a mutual kill). In contrast, it is mentioned in passing that if Vhagar faced Meleys alone, Meleys was big enough that she might have at least held her own. This is why Aegon II made sure to double-team Meleys with both Vhagar and Sunfyre - and even against such unfavorable odds, Meleys still managed to cripple Sunfyre before being killed.
Meleys · Meraxes · Morghul · Seasmoke · Sheepstealer · Shrykos · Silverwing · Sunfyre
Stormcloud · Syrax · Tyraxes · Valryon · Vermax · Vermithor · Vermithrax · Vhagar