What is private equity ? how is it different from venture capital business?

Jeremy J has the right answer. I would only add that Private Equity investors typically play a larger role in managing the company. Because of the large dollars at stake, they take more control of day-day operations in a downturn scenario. Also, the exit options for a Private Equity investment are different from a traditional VC. Since most PE deals are already public companies, PE investors may not have an IPO option to exit the investment. Having said that, it is common for a PE investor to sell (unload) its stake in a company to another PE investor with no additional capital infusion into the company. Another exit option for PE is to split the company and sell the parts for more than the value of the entire company. In some cases, PE firms only invest for this possible outcome. VC's typically don't have this option since the assets of smaller company or a startup is generally not worth more than the value of the entire company.

Venture capital is a form of private equity. A VC firm typically invests in companies much earlier in their life and with limited or no profitablity. There are lots of VC firms that invest in later-stage companies as well and here the definitions blur. Especially as of late, a more traditional PE firm will fund late stage "VC" deals. Typically, however, a PE firm will invest in businesses with stable and consistent earnings.