In these lines, Shakespeare describes the end of the War of the Roses. He tells us that this war (a series of civil wars between the Houses of Lancaster and York, two branches of the Plantagenet) is over, and that the House of York has won. The speaker is Richard, who discusses his brother Edward IV's reign.
The speaker compares the reign of Lancaster and the war to the winter, and the reign of York to spring. In the line "the clouds that lour'd upon our house," the speaker implies that the clouds represent the winter, and the bad times they had just experienced. On the other hand, the word "loured" can be translated as "to frown upon," while "our house" most likely refers to the House of York.