venture capital is an investment meant to help get a promising company off the ground. once the company has some track record of profitability and goes public, the vc's make the bulk of their return, though they often chose to retain some ownership interest. some of the other answers here seem to forget that vc is not angel investing, and does not usually provide seed capital. vc's give the company cash to invest in itself, and the founders a bit of a payoff.
private equity is an investment wherein an investment firm takes a public company private, where it is solely owned by the fund's investors. typically, the goal is to clean up the balance sheet or management, and take the company public again a few years down the line.