The Rains of Castamere immortalized the destruction of House Reyne by Tywin Lannister. House Reyne was obliterated after they rebelled against their liege lord, Tytos Lannister, who was perceived as weak by his own vassals. To restore Lannister dominance, Tytos' son, Tywin, marched against the upstart Lord Reyne. By the end of the rebellion, Castamere had been put to the torch and all members of House Reyne executed. The title is thus a play on words, as the "rains" fall over the empty halls of the "Reynes" who have been killed to the last man. The lyrics heavily reference the fact that the sigil of House Reyne was also a lion, but a red one instead of the golden lion used as the sigil of House Lannister. The rebellion of the Reynes against the Lannisters was thus seen as a clash of lions.
In the decades since young Tywin reasserted House Lannister's dominance by crushing the Reynes, The Rains of Castemere went on to become very popular with soldiers of the Westerlands, becoming an "anthem" of sorts for House Lannister. This extends to the point that even Western soldiers sometimes refer to it simply as, "the Lannister song".
And who are you, the proud lord said, that I must bow so low? Only a cat of a different coat, that's all the truth I know. In a coat of gold or a coat of red, a lion still has claws, And mine are long and sharp, my lord, as long and sharp as yours.
And so he spoke, and so he spoke, that Lord of Castamere, But now the rains weep o'er his hall, with no one there to hear. Yes now the rains weep o'er his hall, and not a soul to hear.