There was one primary critic, today we would call it 'Insentive Problem' or something along those lines. Quite simply, if you give everybody the same amount, why should anybody work hard.

It is the common error of Socialists to overlook the natural indolence of mankind; their tendency to be passive, to be the slaves of habit, to persist indefinitely in a course once chosen. Let them once attain any state of existence which they consider tolerable, and the danger to be apprehended is that they will thenceforth stagnate; will not exert themselves to improve, and by letting their faculties rust, will lose even the energy required to preserve them from deterioration. Competition may not be the best conceivable stimulus, but it is at present a necessary one, and no one can foresee the time when it will not be indispensable to progress. John Stuart Mill [1848]

This critic was basically answered with the believe that in a Socialist/Communist society a 'New Men' would arise, because people adopt to their surrounding:

Man will make it his purpose to master his own feelings, to raise his instincts to the heights of consciousness, to make them transparent, to extend the wires of his will into hidden recesses, and thereby to raise himself to a new plane, to create a higher social biologic type, or, if you please, a superman. Leon Trotsky

The second most important critic, is called 'Economic calculation problem'. This was formulated by Ludwig von Mises in 1920 [1]. Simply stated what Mises did is that he just assumed the socialist were correct about the 'New Men' and went from there. He basically argues that without property rights you are not gone have markets, without markets you don't have prices, and without prices you can't do rational calculation.

This is considered one of the most important debates in the 20th century economics and it would take me to long to get into it. If you are interested in the details I would be happy go into the debate.

[1] Economic Calculation in the Socialist Commonwealth