What is the Marxian definition of corporation? How is it different from the mainstream definition?

There is a world of difference. A capitalist industrial corporation (Marx differentiates that from a merchant orfinancial corporation) is one which gets from its productive employees a value of output that is larger than the total value paid by the corporation for physical inputs (tools, equipment, raw materials) plus the value paid to the workers as wages (payment for what Marx calls their labor power). The difference is the surplus value appropriated by the corporation's board of directors. In the mainstream definition taught in schools (and by me as a professor) there is no such thing as a surplus and hence what I just described is NOT an aspect of the corporation, let alone its central aspect as for Marx. You can see where all this leads by taking a look at an extended discussion in S. Resnick and R. Wolff, Knowledge and Class: A Marxian Critique of Political Economy (Univ of Cgicago Press, 1987), chapter 3.