Alyssa Velaryon was a member of House Velaryon who was married to King Aenys I Targaryen. The marriage was a political match.
Alyssa was the daughter of the master of ships, Lord Aethon Velaryon, and the granddaughter of King Aegon I's master of ships, Lord Daemon Velaryon. She married her cousin, Prince Aenys Targaryen in 22 AC. She gave birth to six children during this marriage: Rhaena, Aegon, Viserys, Jaehaerys, Alysanne and Vaella Targaryen.
Alyssa outlived her husband, and became a hostage to her good-brother, King Maegor I Targaryen, when he took the throne, thereby passing over Aenys' heir, and Alyssa's eldest son, Prince Aegon. Alyssa remained a captive on Dragonstone until the death of Dowager Queen Visenya Targaryen, the chaos of which gave her an opportunity to escape with Jaehaerys and Alysanne, her two youngest still living children. She took the Valyrian sword Dark Sister with her.
Alyssa's escape caused the death of her eldest still living son, Viserys, who was tortured for nine days until he died. His body was was left in the courtyard of the Red Keep, as Maegor hoped Alyssa would come to claim his body. She didn't.
Alyssa rallied lords to the claim of her now only living son, Jaehaerys. Upon Jaehaerys' coronation in 48 AC, Alyssa became Regent. She kept this position until Jaehaerys reached majority two years later. Half a year after stepping down as regent, she married the Hand of the King, Lord Robar Baratheon. With him, she had two more children: Boremund and Jocelyn.