According to Kipling, what was the “White Man's Burden"?

The "White Man's Burden" is the name Rudyard Kipling gave to describe Western imperialists' need to go abroad and colonize foreign nations. For Kipling, this process of colonization is not just about creating business and financial opportunities; it is also about the transmission of Western culture and religion to these nations. This will, in his opinion, makes these inhabitants more civilized.
We see direct evidence of this through Kipling's 1899 poem of the same name, "White Man's Burden," in which he urges imperialists to "send their sons" across the globe to faraway countries. For Kipling, the people of these nations are "half-Devil" and "half-child" and in genuine need of social, cultural and economic redemption. 
For Kipling, this task is burdensome because of its potential risk to the imperialist's reputation. As he says in the poem, many people will not appreciate the enormous efforts involved in carrying out this duty ("thankless years") and, even worse, may cast their judgment negatively:

Cold-edged with dear-bought wisdom, The judgement of your peers.
http://www.kiplingsociety.co.uk/poems_burden.htm