In the books
In A Song of Ice and Fire novels, the Great Spring Sickness is mentioned only once: in A Clash of Kings, Jeor Mormont tells Jon Snow that Valarr Targaryen and his younger brother Matarys, sons of Baelor "Breakspear" Targaryen, who perished earlier at the hands of his brother, died during the Great Spring Sickness; their grandfather died along with them. As a result, the crown passed to Daeron's second son Aerys I.
According to Tales of Dunk and Egg stories and The World of Ice & Fire, the Great Spring Sickness was a plague epidemic that followed the Great Spring of 209 AC, killing tens of thousands in the Seven Kingdoms. A strong man could wake up healthy in the morning and die by the evening. It killed tens of thousands, especially in the major cities; it was bad in Lannisport, worse in Oldtown, but worst of all in King's Landing, where four in ten succumbed to it.
King Daeron the Good, his last Hand of the King, and his grandsons Princes Valarr and Matarys (who had been his immediate heirs after their father died) were both killed during the plague. It also killed the High Septon, a third of the Most Devout, and nearly all of the Silent Sisters (who handle the dead) in King's Landing. Ser Brynden Rivers, the Hand of the King to Daeron's successor, Aerys I Targaryen, ordered the many bodies be brought to the Dragonpit and burned by pyromancers. The light of their wildfire pyres could be seen as a glow throughout the city during the night.
The only regions not affected by the plague were the Vale of Arryn and Dorne. The Arryns closed off the Eastern Road at the Bloody Gate to all travel, as well as blocked all travel by sea, isolating the Vale from the epidemic. Similarly, the deserts of Dorne and the narrow passes of the Red Mountains helped to isolate it from the disease's spread.
The plague swept through Westeros in the intervening years between the first and second of the Tales of Dunk and Egg prequel novellas. Duncan the Tall and his squire Prince Aegon survived due to seeking their fortunes in Dorne at the time it broke out.
In terms of its impact, the Great Spring Sickness is loosely Westeros's analogue of the real-life Black Death, which killed off over a third of the population of Late Medieval Europe.