Commodity: A product manufactured primarily for its exchange value (secondarily for its use value, which it requires only enough to make it sellable). Commodities are fungible, which means they are standardized for the purposes of exchange, and have little or no qualitative difference. Examples: grains, lumber, ipods, fuels, currencies, unskilled labor power, Big Macs.
Surplus value: The social process of capital production, which manifests as the theft of labor power (exploitation) during the conversion of nature (extraction of raw materials) into commodities. Surplus value is the essence and purpose of capitalism.
Surplus value is the portion of exchange value of commodities that is appropriated by the capitalist after underpaying for the labor power it took to make them, plus paying fixed costs such as machinery and inputs. For example, if a worker is paid $2 to produce t-shirts that are sold for $100, and fixed costs associated with their production are $8, then the surplus value is $90.
Capitalists can pay less than labor power is worth because of their private ownership and control over the means of production. They pay less for the raw materials than they are worth, or steal them outright, by conquering and dominating land and other resources (called “primitive accumulation”).
Capital: Re-invested surplus value.
Profit: The revenue left after input costs. It can be the material manifestation of surplus value, or it can be the result of the circulation of capital (for example, through selling services or making unequal trades, buying low and selling high).
Surplus value and profit are not identical. Surplus value can only be generated through commodity production. Profit can be generated in other ways that do not involve commodity production or surplus value. Capitalism rests on surplus value; other forms of profit are considered false or toxic value.