Silverwing was one of the dragons that belonged to House Targaryen. It was once the mount of Queen Alysanne Targaryen. During the great civil war known as the Dance of the Dragons, Silverwing was the mount of Ulf the White.
Silverwing was left riderless on Dragonstone island for many years after the death of Queen Alysanne. During the Dance of the Dragons, Rhaenyra Targaryen's faction - the Blacks - was based on the island, but realized that they possessed more dragons than riders. It was believed that only those of the Targaryen bloodline could successfully bond with dragons, but over the generations the Targaryens had produced numerous bastard children and grandchildren with the commoners of the isle.
Rhaenyra put out a call for any who believed they were a bastard of Targaryen blood to attempt to mount and bond the six riderless dragons, offering reward of gold and title. Many of these bastards - called Dragonseeds - died in the attempt, but ultimately four of them succeeded. Silverwing was claimed by the bastard called Ulf the White.
Hugh and Vermithor fought on Rhaenyra's side in the civil war for some time, but during its later stages Ulf and another dragonseed, Hugh Hammer, decided to betray her. When a large Green army of House Hightower advanced towards King's Landing after a long journey up from the south, Rhaenyra sent an army with two dragons to counter it: Vermithor ridden by Hugh Hammer, and Silverwing ridden by Ulf the White. In the ensuing Sack of Tumbleton, however, both Hugh and Ulf switched sides and turned their dragons against Rhaenyra's own army.
Luckily for Rhaenyra, rather than continue to advance on nearby King's Landing, Hugh and Ulf instead chose to drink and whore amid the ruins of Tumbleton with the Green army, unable to agree on what specific course of action to take next.
In the books
At the start of the Dance, Silverwing was one of the older and larger dragons, apparently one of the powerful second-generation dragons hatched to the Targaryens' original three.
Silverwing's exact coloration hasn't been stated but was presumably silver or silver-white (or at least predominantly silver, it is unknown if she had a secondary color).
Silverwing was famously the mount of Queen Alysanne Targaryen, sister-wife of King Jaehaerys I Targaryen, the Old King whose great and prosperous reign lasted 55 years. He was also a dragon-rider, and his mount was the great Vermithor, largest of the remaining dragons behind Vhagar herself (last of the first-generation dragons). It has been speculated that there is some sort of magical/psychic or at least emotional bond between dragons and their riders, so that they like friends of their rider and scorn their enemies. In this case, Vermithor and Silverwing were actually a mated pair, mirroring the marriage between their two riders, and continued to remain so even after their original riders died. They "oft coiled about each other in the fields" when their armies weren't on the move during the Dance. Their matings may have produced some of the younger dragons at the time of the Dance.
At one point Jaehaerys I and Alysanne made a royal visit to Winterfell, riding Vermithor and Silverwing respectively, along with four other dragons. On a whim, Alysanne rode further up to The Wall itself, and Jaehaerys I followed on Vermithor. They visited the then-headquarters of the Night's Watch, the Nightfort. Seeing how badly it had fallen into disrepair from the dwindling Watch, the royal couple convinced them to abandon it and relocate to the more manageable Castle Black to the east. Some two centuries later, when Samwell Tarly was passing through the Nightfort with Gilly as he returned from beyond the Wall, he thought to himself about how the dragon Silverwing had once stood in that very courtyard, and he idly wondered if she may have left behind any eggs.
The novels never give the lineage of any of the Targaryen dragons, though it is fairly clear that Silverwing was one of the powerful second-generation dragons hatched after the Targaryen Conquest. Live-readings that George R.R. Martin has done of his more extensive private notes mention that towards the end of Aegon I's life, there were about half a dozen dragon hatchlings. When Aegon I died several localized rebellions sprang up against his weak older son Aenys Targaryen, but Maegor helped put them down riding Balerion. Following this, two new dragons hatched at the same time, which was taken as a good omen for relations between the two brothers in the future (which turned out to be false). These two hatchlings may have been Vermithor and Silverwing, given how they were always a pair.
Silverwing was left riderless after Jaehaerys I's death, and had no other until she was claimed by Ulf 30 years later during the Dance. Of the six riderless dragons left on Dragonstone island, she was one of the three who were only semi-wild; compared to the other three who had grown to adulthood without riders at all. As a result, while Silverwing killed a few Dragonseeds who attempted to mount her, she was relatively accustomed to the presence of men and before too long accepted Ulf as her rider.
Ulf rode Silverwing into combat during the great Battle of the Gullet. The Season 5 Histories & Lore featurette recounts events in the wrong order, presenting it as if the loss of Vermax at the Gullet is what prompted Rhaenyra to look for dragonseeds to increase her number of war-capable dragons. In the prequel novellas, however, it was actually the loss of Meleys at the Siege of Rook's Rest which prompted this (the animated featurette greatly condensed this to only briefly mention the Battle of the Gullet in passing). Ulf then rode Silverwing during Rhaenyra's sudden seizure of King's Landing, though the defenseless city fell so quickly that he saw little if any combat.
About half a year later, Hugh and Ulf betrayed against Rhaenyra during the Treason of Tumbleton, turning Vermithor and Silverwing against the Black army and joining the main Green army of the Hightowers which had finally advanced up from the south.
Silverwing was actually the only adult rideable dragon to survive the Dance. Only four dragons were left alive at the end of the war including her, though one of them (Morning) was hatched late in the war, and the other two (Sheepstealer and the Cannibal) simply vanished never to be seen again. Silverwing, however, went mad with grief over the death of Vermithor and all of her other cousins in battle. After Ulf was killed in the war she would accept no new rider but devoured any man who approached her, then fled. She nested around Red Lake in the northwest of the Reach for some years, hunting and killing as she would. Through as yet unclear circumstances she must have died during the next two decades because it is definitively stated that all remaining dragons were dead by the end of the reign of Aegon III.
Meleys · Meraxes · Morghul · Seasmoke · Sheepstealer · Shrykos · Silverwing · Sunfyre
Stormcloud · Syrax · Tyraxes · Valryon · Vermax · Vermithor · Vermithrax · Vhagar