The weakness of all Utopias is this, that they take the greatest difficulty of man and assume it to be overcome, and then give an elaborate account of the overcoming of the smaller ones.
-C. K. Chesterton, Heretics  

George Orwell dealt with the disadvantages of Communism in his allegorical book Animal Farm (1945). In that story not only pigs are selfish and greedy. The pigs are just smarter and more practical and ruthless about taking care of their own interests. Almost all the other animals are selfish and greedy except for Boxer the workhorse. This is the greatest difficulty of man. People won't work for the common good. People won't cooperate. People won't share. Boxer is an exception--and we see how he is exploited.

As Chesterton says, this is the weakness of all Utopias. It leads to the application of force, and this only undermines initiative and creates resentment and greater disharmony. When the Germans invaded the Soviet Union in 1941 they were welcomed as deliverers by many of the peasants who thought they were going to regain their independence. Then they discovered that the Germans were just as bad as their Bolshevik rulers, if not worse. 

B. F. Skinner wrote a utopian book titled Walden Two (1948). His socialistic ideas seemed workable because he envisioned building an ideal society on a much smaller scale as a sort of prototype. But when a group of young men and women tried to put them into practice during the period when communes were all the rage in America, they found that some people did all the work and a lot of the others were lazy, incompetent, uncooperative, and parasitical. People who are intelligent, ambitious, creative, industrious, and gifted don't want to be tied to a lot of others who have little or nothing to offer. When Fidel Castro took power in 1959, many of the intelligent, educated professional people just wanted to get out of Cuba rather than live under a dictatorship. 

The main advantage of communist and socialist schemes is that if everybody worked together nobody would have to work very hard and everybody could be comfortable. The usual result has been, as illustrated in Orwell's Animal Farm, that power is transferred from one group, the capitalists, to another group, the party leaders, and you have the same thing in a different guise. The Soviets eventually had to acknowledge that top-down management just wasn't working and that a return to private enterprise was the only hope of getting out of the mess they had created. This happened very spectacularly under Mikhail Gorbachev and Boris Yeltsin in the 1980s and 1990s.

If communism worked the way it is supposed to in theory, the main advantage would be that it would be a fair system.  People would not have to work hard, only to remain poor.  They would not have others get rich off of their hard work.

The major disadvantage of communism is that it doesn't work.  Communism relies on the ability of government to plan an economy.  But an economy is too big to be centrally planned.  Communism relies on people to work hard and do their absolute best without any tangible incentive to do so.  It does not appeal to people's selfish desires the way capitalism does.  For these reasons and others, it has never worked the way it is supposed to.