Payment-processing giant First Data Corp. is returning to the public market nearly a decade after it was taken private in one of the biggest leveraged buyouts on record.

First Data, which handles trillions of dollars of transactions such as credit card payments each year, was taken private by KKR and its investors in 2007 in a $26 billion deal. It was one of the largest private-equity takeovers of a public company in history at the time, done during the peak for megabuyouts before the financial crisis in 2008.