Well, basically the market allocates resources towards the most urgent uses.
How does the market do the job?
Through prices. Imagine money as votes, and prices reflecting the number of votes each resource(either capital or consumer goods) gets. The market place is where people bid for resources with money votes. So, basically the market is about, as already mentioned, allocating scarce resources towards the most urgent uses.
So, what’s the calculation problem?
Without a pricing system, there is no scope of any rational allocation of resources, that is, towards the most urgent uses(based on the number of votes each resource gets).
This is the most basic explanation of the calculation problem you need to know.
Next, there were some of these neo-classicals who started conceding that without any pricing system rational allocation of resources would be impossible. So, they said that the pricing mechanism would still be intact without private property. But, they did not concede the disadvantages the lack of private property rights would bring to their system. The neo-classicals proposed a system in which pricing mechanism does exist, but it would be entirely controlled by the State.
What was this neo-classical pricing system like?
I basically consider two important neo-classical economists to be worthy of rebuttal. One was Oscar Lange, and the other was Enrico Barone.
Oscar Lange basically accepted that people should vote in the market place with their money votes, and prices of consumer goods should be ascertained. But Lange was not able to solve the problem of pricing of capital goods. Why? In a market economy, multiple individuals or companies own capital resources(like land, mines, etc.). These individuals bid in the capital goods market to ascertain the prices of these capital goods. But, in the socialist economy where there is just a single owner of capital resources. So there is no question of competitive bidding happening. The best that the planner could do would be to impute values to capital goods based on the price of the consumer goods that are made out of it. This imputation process is quite impossible. There are so many components that go into making a consumer good, and ascertaining the price of each component is not just impossible. The Capitalist economy on the other hand has decentralized ownership, which makes it possible for the economy to price individual components in the market place.
Enrico Barone on the other hand did not want any type of genuine pricing. His model for ascertaining prices basically assumes that the State has all knowledge that it wants to price commodities. The State needs to have the consumer preferences, the current technological capabilities of the economy, the amount of capital goods in the economy’s stock, etc, etc.. And when the State has all this required information, prices can be ascertained. Sure, it can be. If God helped us!
The other solutions
There have been other solutions to the problem. And mostof these solutions are just ways to sneak through a rational pricing system but with the same prejudice against private property rights. Some guys proposed that there would be competing branches of the government which would bid for capital resources, and thereby ascertain their prices. The disadvantage of such a system is obvious. There is no surety that the monopolies will be efficient, for they are appointed by their political masters. The State still owns the capital resources, the monopolies just bid. So the State can still make arbitrary allocations. There is no guarantee that the monopolies have all the knowledge of techniques that could be used to make products. In a Capitalist economy an entrepreneur who thinks he can do better than others does not have to petition the State to get to apply his ideas. Things would be quite pathetic in a Socialist State. And I can go on!