This is probably more of an "anthology of quotes" than an original explanation, but I think the quotes speak for themselves quite well ;)
Industrial capital is basically what Marx covered extensively in volume 1 of Das Kapital.
It's what you usually think about when you hear the word "capital"...either the physical means of production themselves or the means to acquire them.
Finance capital is a little different. If I am not mistaken the term was first coined by Hilferding, and he explains what he means by it best in my opinion:
Originally posted by Hilferding+--> (Hilferding)We have seen how in the beginning of capitalist production the money of the banks comes from two sources. First, from the money of the non-producing classes; secondly, from the reserve capital of the industrial and commercial capitalists. We have seen, further, how the development of credit tends to place at the disposal of industry not only the whole reserve capital of the capitalist class, but also the greatest part of the money of the unproductive classes. Present-day industry, in other words, is carried on by means of a capital far larger than the total capital in the possession of the industrial capitalists. With capitalist development the sum of money constantly grows which is placed by the non-producing class at the disposal of the banks, and by these latter at the disposal of industry. The disposal over these sums, so indispensable to industry, belongs to the banks. With the development of capitalism and its credit organisations there thus grows the dependence of industry upon the banks. On the other hand, the banks can only draw the moneys of the non-productive classes, and keep the ever-increasing foundation stock of the same at their permanent disposal by paying interest on these moneys. This they could do, as long as these sums were not too extensive, by making use of them for speculation credit and circulation credit. With the growth of these sums on the one hand, and, on the other, with the decreasing importance of speculation and commerce, it became necessary to convert them more and more into industrial capital. Without the steady extension of production credit the possibility of making use of the deposits, and therewith also the paying of interest on the bank deposits, would long ago have sunk much lower. This is partially the case in England, where the deposit banks only negotiate circulation credit, the interest on the deposit being therefore only minimal. Hence the continual departure of the deposits into spheres of industrial investment by the purchase of shares. Here the public does directly what, in the case the founder’s profit does not come to them. But for industry, it means less dependence on bank-capital in England in comparison with Germany.
The dependence of industry on the banks is thus the result of the conditions of property. An ever-increasing portion of the industrial does not belong to the industrials who use it. They only receive the disposal over it from the bank, which, as far as they are concerned, represents the owner. On the other hand, the bank has to fix an ever growing portion of its capital in industry. It therefore, becomes, in an ever-growing measure, an industrial capitalist. I call this bank-capital – that is, capital in money form – which in this way is converted in reality into industrial-capital, the finance capital. Towards the owners it always conserves its money-form, is invested by them in the form of money-capital, and can at any time be withdrawn by them in money form. But in reality the greater part of the capital thus invested in the banks is converted into industrial, productive capital (means of production and labour-power) and fixed in the process of production. An ever greater portion of the capital employed in industry is finance-capital, capital at the disposal of the banks, and being made use of by the industrials.
The finance-capital develops with the development of the joint-stock companies and reaches its height with the monopolisation of industry. The industrial revenue becomes a steady and increasing one. Thus the power of the bank-capital to invest in industry gains ever further extension. But the bank-capital is at the disposal of the bank, and the bank is ruled by the owners of the majority of the bank shares. It is clear that with the increasing concentration of property the owners of the fictive capital, which gives power over the banks, and of that which gives power over industry, are becoming more and more identical. All the more, in that, as we have seen, the large banks are ever gaining more and more power of disposal over the fictive capital.
Though we have seen how industry is becoming more and more dependent upon bank-capital, that by no means involves the industrial magnates. Just as, on the contrary, capital itself, on reaching its highest stage, becomes finance-capital, so the magnate of capital, the finance capitalist, comes more and more to unite the disposal over the total national capital by ruling over the bank-capital. Here, too, the personal union plays an important part.
With cartellisation and trustification, finance-capital reaches the highest stage of its power, while the commercial capital experiences its deepest degradation.[/b] --emphasis added
From Hilferding's Finance Capital which, as far as I know, is unavailable online.
By virtue of the law of accumulation, and supposing the bold section of the text is correct, an economy that's capitalist will "inevitably" have finance capital.
Actually, Engels wrote quite a bit on the concept of "joint stock" companies, which is interesting in and of itself:
Originally posted by Engels+--> (Engels)This rebellion of the productive forces, as they grow more and more powerful, against their quality as capital, this stronger and stronger command that their social character shall be recognised, forces the capitalist class itself to treat them more and more as social productive forces, so far as this is possible under capitalist conditions. The period of industrial high pressure, with its unbounded inflation of credit, not less than the crash itself, by the collapse of great capitalist establishments, tends to bring about that form of the socialisation of great masses of means of production which we meet with in the different kinds of joint-stock companies. Many of these means of production and of communication are, from the outset, so colossal that, like the railways, they exclude all other forms of capitalistic exploitation. At a further stage of evolution this form also becomes insufficient: the official representative of capitalist society — the state — will ultimately have to *10 undertake the direction of production. This necessity for conversion into state property is felt first in the great institutions for intercourse and communication — the post office, the telegraphs, the railways.
If the crises demonstrate the incapacity of the bourgeoisie for managing any longer modern productive forces, the transformation of the great establishments for production and distribution into joint-stock companies and state property shows how unnecessary the bourgeoisie are for that purpose. All the social functions of the capitalist are now performed by salaried employees. The capitalist has no further social function than that of pocketing dividends, tearing off coupons, and gambling on the Stock Exchange, where the different capitalists despoil one another of their capital. At first the capitalist mode of production forces out the workers. Now it forces out the capitalists, and reduces them, just as it reduced the workers, to the ranks of the surplus population, although not immediately into those of the industrial reserve army.
But the transformation, either into joint-stock companies, or into state ownership, does not do away with the capitalistic nature of the productive forces. In the joint-stock companies this is obvious. And the modern state, again, is only the organisation that bourgeois society takes on in order to support the general external conditions of the capitalist mode of production against the encroachments as well of the workers as of individual capitalists. The modern state, no matter what its form, is essentially a capitalist machine, the state of the capitalists, the ideal personification of the total national capital. The more it proceeds to the taking over of productive forces, the more does it actually become the national capitalist, the more citizens does it exploit. The workers remain wage-workers — proletarians. The capitalist relation is not done away with. It is rather brought to a head. But, brought to a head, it topples over. State ownership of the productive forces is not the solution of the conflict, but concealed within it are the technical conditions that form the elements of that solution.[/b] Anti-Duhring Chapter 24 "Theoretical Socialism" (http://www.marxists.org/archive/marx/works/1877/anti-duhring/ch24.htm) by Frederick Engels (1877).
There's a similar passage written by Engels in Socialism: Utopian and Scientific Chapter 3 "Historical Materialism" (http://www.marxists.org/archive/marx/works/1880/soc-utop/ch03.htm).
Karl Marx uses the term less frequently, but nonetheless still interesting to read:
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The railways sprang up first as the couronnement de l'oeuvre in those countries where modern industry was most developed, England, United States, Belgium, France, etc. I call them the "couronnement de l'oeuvre" not only in the sense that they were at last (together with steamships for oceanic intercourse and the telegraphs) the means of communication adequate to the modern means of production, but also in so far as they were the basis of immense joint stock companies, forming at the same time a new starting point for all other sorts of joint stock companies, to commence by banking companies. They gave in one word, an impetus never before suspected to the concentration of capital, and also to the accelerated and immensely enlarged cosmopolitan activity of loanable capital, thus embracing the whole world in a network of financial swindling and mutual indebtedness, the capitalist form of "international" brotherhood. Letter to Nikolai Danielson (http://www.marxists.org/archive/marx/works/1879/letters/79_04_10.htm) by Karl Marx (1879).
It's a logical conclusion from Marx and Engels' writings, it's not something really pulled out of left field (so to speak).
Lenin also quotes Hilferding, which may be useful too in considering what is financial capital:
Lenin
“A steadily increasing proportion of capital in industry,” writes Hilferding, “ceases to belong to the industrialists who employ it. They obtain the use of it only through the medium of the banks which, in relation to them, represent the owners of the capital. On the other hand, the bank is forced to sink an increasing share of its funds in industry. Thus, to an ever greater degree the banker is being transformed into an industrial capitalist. This bank capital, i.e., capital in money form, which is thus actually transformed into industrial capital, I call ‘finance capital’.” “Finance capital is capital controlled by banks and employed by industrialists.” Imperialism: The Highest Stage of Capitalism Chapter 3 (http://www.marxists.org/archive/lenin/works/1916/imp-hsc/ch03.htm) (1916).
It's finance capital that's introduced to contrast itself with "regular, old fashioned" industrial capital.