In the aftermath of the American Revolution, the British government was sensitive to unrest in its remaining colonies with large populations of British colonists. After William Lyon Mackenzie's abortive Upper Canada Rebellion in 1837 and Louis-Joseph Papineau's matching Patriotes Rebellion in Lower Canada that lasted through the next year, Lord Durham was appointed governor general of Upper Canada (now Ontario) and given the task of examing the issues and determining how to defuse tensions. In his report, one of his recommendations was that colonies which were sufficiently developed should be granted "responsible government", a term which specifically meant the policy of British-appointed governors bowing to the will of elected colonial assemblies. It is worth noting that this is, even to this day, de facto rather than de jure in some former British colonies.
To take a specific example, the Governor-General of Canada, theoretically appointed by the Queen of Canada, has the power to veto any and all legislation passed by the Canadian Parliament. In practice, however, any breach of the policy of responsible government would undoubtedly touch off a constitutional crisis. (See Queen of Canada for more information on the Canadian monarchy.)
In time, the granting of responsible government became the first step on the road to complete independence. In contrast to the American experience, Canada (for example), gradually gained greater and greater autonomy over a considerable period of time (including 1867's British North America Act, 1931's Statute of Westminster, and even as late as the repatriation of the British North America Act in 1982).