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Income redistribution, in its most basic form, is the taking of income from one person to be given to someone else. This could be anything from the complete confiscation of a person's income to give to a lazy kind (seen in no current Western nation), to a tax on the vast majority in order to support a minority of completely disabled people.
 Modern welfare
The modern welfare state is based, to some extent, on a modest amount of income redistribution. In the United States, the popular Social Security program exists as a modest income redistribution program, giving money from workers to widows, retirees, and the disabled.
When presented this way, most people have no issues with income redistribution. However, many conservatives and libertarians oppose any form of income redistribution as theft. One set of reasoning is that if one man takes money from you he is a thief, if a hundred men take money from you they are a crime family, and if ten thousand men take money from you they are a government. This is coupled with the belief that the morals of taking money do not change depending on the number of people doing the taking (from one thief to a ten thousand strong government).
The other rationalisation of the opposition to income redistribution is that it is punishing people for being productive and earning money while rewarding those who are not productive. (With "productive" meaning "playing golf while some other people do all the actual work," presumably.)