Definition
a duty formerly asserted by white people to manage the affairs of nonwhite people whom they believed to be less developed
This phrase comes from a poem (by the same name) by the imperialist poet Rudyard Kipling. It was about the Philippine American War, where Kipling encourages the American annexation and colonisation of Philippine Islands.

an advertisement of Pears Soap which talks about White Manâs burden and how teaching cleanliness will fulfill it.
Full poem
Take up the White Manâs burdenâ     Send forth the best ye breedâ Go bind your sons to exile     To serve your captivesâ need; To wait in heavy harness     On fluttered folk and wildâ Your new-caught, sullen peoples,     Half devil and half child.
Take up the White Manâs burdenâ     In patience to abide, To veil the threat of terror     And check the show of pride; By open speech and simple,     An hundred times made plain. To seek anotherâs profit,     And work anotherâs gain.
Take up the White Manâs burdenâ     The savage wars of peaceâ Fill full the mouth of Famine     And bid the sickness cease; And when your goal is nearest     The end for others sought, Watch Sloth and heathen Folly     Bring all your hopes to nought.
Take up the White Manâs burdenâ Â Â Â Â No tawdry rule of kings, But toil of serf and sweeperâ Â Â Â Â The tale of common things. The ports ye shall not enter, Â Â Â Â The roads ye shall not tread, Go make them with your living, Â Â Â Â And mark them with your dead!
Take up the White Manâs burdenâ     And reap his old reward: The blame of those ye better,     The hate of those ye guardâ The cry of hosts ye humour     (Ah, slowly!) toward the light:â âWhy brought ye us from bondage,     Our loved Egyptian night?â
Take up the White Manâs burdenâ     Ye dare not stoop to less Nor call too loud on Freedom     To cloak your weariness; By all ye cry or whisper,     By all ye leave or do, The silent, sullen peoples     Shall weigh your Gods and you.
Take up the White Manâs burdenâ     Have done with childish daysâ The lightly proffered laurel,     The easy, ungrudged praise. Comes now, to search your manhood     Through all the thankless years, Cold-edged with dear-bought wisdom,     The judgment of your peers!