The Old Way refers to the ancient Ironborn tradition of reaving and plunder. A key concept within the Old Way is paying the "iron price", which means to take possessions from defeated adversaries, rather than purchasing or bartering for them, which is referred to as the "gold price."
Before the coming of Aegon the Conqueror, the Ironborn carved out a vast domain in Westeros through lightning raids, carrying away plunder, thralls and women, and burning everything else. The Old Way ended after Aegon the Conqueror burned Black Harren and his sons in Harrenhal and vanquished the Ironborn. Aegon outlawed the Ironborn's reaving ways, at least in Westeros. The Ironborn were still free to reave and plunder the Free Cities of Essos, preventing the Old Way from complete extinction.
Ever since the Conquest, several Ironborn rulers have made attempts at returning to the Old Way, and striking at the rest of Westeros. While these attempts have been momentarily successful, they have always ended in defeat.