Cambridge Capital Controversy for Laymen

I haven't read the original papers, only summaries in Sraffa-influenced literature. The basic idea concerns the definition of capital and the measurement of the quantity of capital. The problems at this level have repercussions at higher levels. These are known as reswitching and capital-reversing (these were what was mostly discussed, at least as far as I can tell). To learn about the latter two, you'll have to check out some of the formal literature or Wikipedia. Here's the gist of the basic problem, very informally:

One can view "capital" both as quantities of physical goods (machinery, raw material, buildings etc.) and as a sum of values (prices) expressed in money. Think of the use-value and value distinction in Marx. In the former case, a given capital consists of quantities of qualitatively heterogeneous "stuff", e.g., "1 spinning jenny and 10 pounds of cotton". There is no way to aggregate this meaningfully and say that this given capital is x homogeneous units. Any aggregation from this physical point of view is like counting apples and oranges. However, if we choose to measure capital in values (prices), then if the jenny costs £x and 1 pound of cotton costs £y, one should be able to somehow sum over this, as it's all denominated in the same unit (£).

The problem is that the latter option (measuring capital in values), as derived from the presuppositions of neoclassical economics, leads to circularity: to determine the price (i.e., the quantity) of capital, you need to determine the rate of profit (here the idea is that the price of capital depends on how much profit it would turn out if it were put to use; the ratio of this "how much" is precisely the rate of profit), but to determine the rate of profit, you need to know the price of capital (because the rate of profit depends on the quantity of capital: if a capitalist makes £10k in profit, then his rate of profit will be different depending on whether he'd invested £5k, £10k or £20k). Alternatively, you could decide to determine the quantity of capital by the cost of production of 1 spinning jenny and 10 pounds cotton, but in neoclassical economics, this route also involves the rate of profit, so the circularity remains.

However, it can be made to work under some assumptions, like a single-sector economy with a single capital good (like a basic corn model where corn is combined with labor to grow corn as a subsistence good and as a capital good). The problem is that such an economy is nothing like a real capitalist economy. Thus neoclassical economics provides no theory of a capitalist economy, and when you actually try to apply it to a capitalist economy, you end up with circular definitions.