The basic answer is that it is a great idea on paper, but a lousy practical application.
Communism as such is a broad term to encompass the general ideology of a socio-economic equality. In its school of thought the world order stems from an economic system: basically it is divided into the haves and the havenots. It argues that a more equal system could be established via social ownership -- what's yours is mine and what's mine is yours.
While theoretically a noble idea that would in practice abolish the poverty gap, racial and economic discrimination, it has yet to see a working model. Countries that have adopted this idea have had varying degrees of success. While a certain 'equality' was established, it is within human nature to create social barriers. Suddenly it was The State that ruled as opposed to the people and it was The State that termined individual privileges within the system, regardless of individuality.
Countries now (red) under nominally Marxist-Leninist governments and previously (orange)
With the government becoming the central power over everything and everyone, Communism would once again shift to an unequal paradigm: dividing the world into those who are in on the system and those who are out. Who was who was once again determined by the very system and whoever ran it, opening up massive gaps of inequality that resulted in even bigger socio-economic gaps within the population at the cost of individuality.
In other words: cool idea, it just doesn't work.
Read on: Socialism: Utopian and Scientific
A Theory Of Socialism and Capitalism