The major weakness of Marxism is that it does not seem to work in the real world. This is because it does not take into account the essential greediness and selfishness of the human being. Marxism relies on people to work hard just because they should and to forego the ability to get rich from their efforts. Unfortunately, in the real world, this does not happen and Marxist economies have always been inefficient and unproductive.
The major strength of Marxism is a theoretical one. In theory, it would be a much more humane way to run a society. It would be much better if we could have societies where no one oppressed any one else and where no one was poor. This is what Marxism promises (though it has never been able to deliver it).
Marxism is designed to do away with poverty--the state controls the means of production and allocates these according to one's needs. Everyone is supposed to work hard and contribute to society. This is supposed to be a type of utopia. During the 1930s Marxism grew in popularity in the United States because the Soviet Union had such low unemployment numbers.
In reality, Marxism has the capacity to be quite corrupt. This is in part because so-called "Marxist" states usually violate key principals of Marxism. When states have tried to implement a version of Marxism, bureaucrats often take more than their fair share while the people suffer. Additionally, without incentive, people lose their willingness to go "above and beyond" to get a job done. People might do shoddy work in order to meet mandated quotas of production. Without individual incentive to improve, economic life could stagnate under Marxism. Marxism is designed to create a utopia, but it fails to take human nature into account and one of it's greatest weaknesses is that it seems impossible to truly implement.
If you are looking for a demonstration of how marxism can go wrong, a suggested book is Animal Farm. This will explain mainly the bad points of marxism.
One of the good points, that I think is a great feature of marxism, is that it encourages all people to believe, dream, and ultimately, work to create a Utopian society.
However, is it right to allow people to dream such things, when, as animal farm shows,they can never be achieved?