An essential component of communism is the idea of "no private property". But this is simply a label, not a complete ideological or political position. We have to look at the issue more closely.
First, the idea of "no private property" is not best understood regarding the ordinary physical property that people typically own (or control in conditions similar to simple ownership): clothes, furniture, dwellings, cars, etc. The communist trope goes, "Communism doesn't mean we all have to share a toothbrush."
On the other hand, I've actually lived in a commune with no private physical property literally beyond my toothbrush and my underwear. And it was far and away the highest material standard of living I've ever enjoyed (even counting my current DINK lifestyle) despite the fact that only about half the commune was working in "straight" jobs and it was during the 80's recession. Not only did we have a high standard of living, we also had the capital to start several businesses and buy a house in San Francisco. So the idea of no private physical property is plausible.
The primary focus of the communist idea of no private property regards absentee ownership, such as a landlord renting a house, investors owning a factory, or bankers owning money. Specifically situations where people own things without having any kind of physical relationship to the property. For example, I physically possess my house, but my landlord owns it... at least on paper.
Fundamentally, the idea of abstract, absentee ownership alienates the work required to produce goods and services from the rewards of efficiently producing them. If a worker becomes more efficient, the rewards of that efficiency go not to himself but to the owner of capital. The only way, then, to motivate a worker on a long term basis is to keep him in conditions of "work (faster) or starve."
This fundamental long-term problem is masked to a certain degree in an expanding economy, because an expanding economy is chaotic and has local pockets of labor shortages. It is the local shortage, not the efficiency of the labor, that drives up labor prices. But, because labor is commoditized, the higher labor prices driven by the local shortage attracts more labor to the "pocket", driving labor prices down to the minimum cost necessary to keep the laborers alive and productive.
All markets, free or otherwise, always drive resources to the bottlenecks: i.e. those places in the movement and transformation of raw materials to commodities where demand exceeds supply. All markets either force prices down to the labor cost, or they drive prices up to the actual subjective use-value.
What capitalism does, by establishing and coercively enforcing absentee ownership (how else would you establish absentee ownership but with coercion?) is make ownership of capital a structural bottleneck, a bottleneck that cannot be resolved by adding more resources to it. It's a pure positive feedback system, with no corresponding negative feedback to prevent overload. It doesn't take a conspiracy or cooperation to make all the surplus resources go to the owners of capital, it's simply a structural feature of capitalism.
This structure violates our basic moral intuitions about fairness and deserts: Why should someone get more for no better reason than simply because they presently have more? It takes no skill, talent or work to use capital to make more capital; indeed capitalists actually boast about letting their money work for them. And when this system is enforced by actually making people starve or live in conditions of desperate poverty, we must engage in massive denial to uphold the system.
But there's not just a moral issue: there are pragmatic issues as well.
Because the positive feedback mechanism that causes the one-way accumulation of capital is structural, not planned, capitalists do not acquire experience or build trust to work together on a long-term basis. The competition within the capitalist class is fierce, pitiless and unrelenting. (Even so, this intra-capitalist competition does not act as a negative feedback system, because it merely moves resources around within the capitalist class; it moves only trivial resources back to the working classes.) No matter how many real, physical resources the capitalists acquire, there's always an enormous incentive to cut costs.
Since labor is commoditized, in addition to making production more efficient in a real sense, by reducing the socially necessary labor time to produce a given commodity, there's an additional incentive under capitalism to make production more efficient in a financial sense, by reducing the surplus resources allocated to the actual workers in exchange for their labor. Unless there's a temporary, local labor shortage, you simply cannot survive as a capitalist unless you do everything you can to pay your workers as little as necessary to keep them alive and productive, and spend nothing at all on their survival once they have aged past peak performance.
Furthermore, because capitalism intrinsically alienates the actual work from reward, it's psychologically and socially easy to alienate it a step further: to have money represent not physical productive capability, but financial capability; to remove the annoying step of actually creating commodities to transform money into more money. This second level of alienation is precisely what we see in the present global financial crisis. With the pure Ponzi schemes finally unraveling, the ruling capitalist class is simply paralyzed. And when the Gods fail, it is the humans who suffer.
Human nature being what it is, these pressures inevitably ends with the capitalists pushing the workers' standard of living so low that they have nothing to lose by open rebellion. It's not just a communist-driven phenomenon: we see these slave, peasant, worker and welfare-class rebellions in every society (cough Spartacus) where a ruling class commoditizes and dominates a working class.
In the past, no military technology was able to withstand a deep-rooted working-class rebellion. A million starving peasants could just spit on the king's army and drown them. Today, perhaps, not so much: with biological, chemical and nuclear weapons, and an army willing to use them unquestioningly on their fellow human beings (even on their own countrymen!), it's entirely possible that the capitalist class, when pushed to the wall, will simply engage in genocide and mass murder on a scale that will make Hitler look like a humanitarian.
The cure for these problems is socialism. And not just any old kind of socialism, but socialism that intentionally and explicitly acts as a transition to communism: an end to all class relations, an end to all economic and political exploitation, and an egalitarian society sharing the abundance of material wealth we have the capacity to produce today.
Socialism starts with the state directly being the only (or the dominant) absentee owner: Ownership of dwellings, ownership of capital and (as we're doing right now) ownership of money.
You might object, "But I don't want the state to own the capital!" Too late, it already does. More precisely, the owners of capital own the state: our present government exists primarily to protect the interests and private property of the owners of capital. Any sop to the working class results only from dimly enlightened capitalists (i.e. Democrats) who understand the danger of open rebellion. But even now, with the economy in ruins, a quagmire in Iraq, and eight years of the stupidest president and ruling party since... well... ever, the Democratic party still cannot garner 60% of the vote, and they're rushing as fast as (or faster than) the Republicans to transfer real material wealth from the working class (taxpayers) to the capitalist class (bankers)... to prop up the system; to rescue capitalism and the capitalist class from their own incompetence and egregious stupidity.
Regulatory/welfare capitalism, a la Franklin Roosevelt and Lyndon Johnson (and to some extent Richard Nixon, surprisingly enough) is a good idea, at least on paper. Use the government to set a "floor" for financial labor costs, thus ensuring that labor commoditization does not drive the price of labor to the bottom, to ensure that at least some of the surplus value of labor goes directly to the laborers. It actually worked, kinda sorta, for 40+ years.
But regulatory capitalism is inherently unstable, because it leaves too much power in the hands of the capitalists, and the vicious intra-captialist competition means that every capitalist has an enormous incentive to find some way around the regulations... especially if he can ensure that the regulations still apply to his competition. All you need is one exception (cough Mariana Islands) and the whole system starts to unravel. Add that to the obvious strategy of using money to dominate the mass media, and as we have seen in present-day history we see that the power of capital overwhelms the power of an uncoordinated, isolated and alienated population.
The problem is that capitalism makes ownership, especially absentee ownership, completely unaccountable to the public interest. We simply cannot take away Bill Gates money in anything even resembling the same way we can take away George W. Bush's political power as President of the United States. Next January, Bush will simply be just another citizen, but Bill Gates will always have the economic power to decide what is valuable. This power is inalienable: No matter how incompetently he tries to create value (cough Windows Vista), he will keep his money.
The problem of keeping a socialist state accountable to the people — without a nongovernmental capitalist class acting as something of a counterweight — is presently unsolved. But we know that having a capitalist class with an inalienable right to absentee ownership provides minimal accountability, and they eventually erode what little accountability that can be established politically.
It's a scary thing: I'm proposing abandoning the devil we know, capitalism, for the devil we don't know, socialism. But I don't think we have much choice: The devil we know has every incentive to kill most of us and enslave the rest. I don't see how socialism could be worse.