Contrary to what the ruling class would have us believe, capital is not just a large amount of money. If you win the lotto, that does not mean you are a successful capitalist.
Capital is not just an amount of money, machinery or equipment. It is also a social relationship. It doesn't just grow on trees, it must be produced. It is accumulated labor (Marx also called it dead labor). But this accumulation of labor is necessary for any system of production - it only becomes capital with certain social relations.
Accumulated labor can only become capital when it can be exchanged for living labor for the purpose of expanding capital. For capitalism to exist, there must be a class of people separated from the means of production. Because they do not own or control the means of production and therefore, the means of survival, they must work for those who do. Capital therefore implies wage-labor; they are two sides to the same coin.
Also, capital can only exist as many capitals. Each capital must produce in competition with the other or face destruction. It is this competition which compels those who posses accumulated labor to use it as capital, to employ workers, who thereby expand its value, rather than just consuming it themselves. Ford's executives are not driven by personal greed alone to make more and more profits, they must compete with GM, Chrysler and Toyota.
Production for production's sake, accumulation for accumulation's sake; this is the basic dynamic of capitalism.
The question of private ownership is of secondary importance. As long as workers are separated from the means of production and are compelled to work for the tiny minority that control it, and as long as the different tiny minorities are in competition with one another, you still have capital and capitalism. It doesn't matter whether one works for Nike or the USSR's government. Both are driven by accumulation for accumulation's sake.
Capitalism is a system in which living labor is dominated by dead labor, in which living labor exists wholly and soley to increase accumulated labor. The worker becomes nothing more than an appendage of the machine, a mere production cost to be replaced the second it "malfunctions".
It is a system based on for profits and not people.