Dragon

Vhagar

Queen of All Dragons

House: Targaryen
Standing: Alive at the season 3 cutoff Alive at the cutoff. Her rider is poisoned in season 3, episode 7 and his fate is not shown, so who mounts her next is open.
Riders: Visenya Targaryen, Laena Velaryon, Aemond Targaryen
Size: The oldest and largest living dragon, a hundred and eighty years old and the size of a small hill. Nothing in the sky can fight her and win.
On screen: S1E6 to S3E8

The single most decisive thing either side owns. Every strategic conversation in seasons 2 and 3 is really a conversation about where Vhagar is.

Where it turns up

S1E7 Aemond claims her at Driftmark and loses an eye to Lucerys in the fight afterwards.
S1E10 Kills Arrax and Lucerys Velaryon in the storm over Storm's End, past the point where Aemond can call her off.
S2E4 Kills Meleys at Rook's Rest and cripples Sunfyre with Aegon still on him.
S3E2 Carries a wounded Aemond to Harrenhal, where he kills Simon Strong and then collapses into Alys Rivers's care.
S3E7 Rhaena plots to kill her in her sleep. Daemon stops it.
S3E8 Aegon urges his brother to get back on her and take the throne. Whether he does is not shown.