This one comes up so often that there's a whole section in the FAQ about it.

Heavily paraphrasing /u/sowser 's response (which you should go read): New World slavery was inherently racially based. Any reference you may have heard to 'white slavery' is probably referring to indentured servitude which is different because it is temporary, it arose from mutual agreements, and the person was not considered property.

I want to say this at the outset, because I think it's significant from a historiographic standpoint, but you are most likely to encounter the concept of "white slavery" in the context of contemporary White Power rhetoric, especially as means to delegitimize the significance and uniqueness of the harm caused by European and American chattel slavery targeting African-descended peoples.

The most commonly cited example of "white slavery" occurred at the hands of the Barbary raiders in the late 18th and early 19th centuries. What occurred there was much closer to the military practice known as "impressment," where men were kidnapped from merchant vessels and forced to work on board military vessels. In the case of the Barbary raiders, this meant that white Europeans and Americans were sometimes forced to work aboard quasi-military vessels belonging to the various Barbary states. In the vast majority of cases this impressment did not constitute chattel slavery, as the men were often allowed to integrate into North African society once their raider landed. There are cases in the historical record that indicate that it was not uncommon for European and American captives to convert to Islam by choice and continue to live in Africa in post-conflict periods (variously between European powers/United States and the Barbary states). To be clear: it is highly likely that white Europeans and Americans were literally enslaved, but the cases are extremely limited compared to other enslavement regimes throughout history.

That said, American popular culture did use the term "white slavery" in the early nineteenth century to describe Barbary impressment, although my survey of the primary source literature in the period suggests that "captivity" rather than "slavery" was the descriptor of choice (NB: this is unpublished research for my dissertation, but I thought I would include it nonetheless despite the possible rules violation). It is important to note that anti-Barbary propaganda was rife in the United States before 1817, in part because of Jeffersonian efforts to build support for the first Barbary War (Carol Winkler has done interesting research on the Reagan administration's use of Jefferson's tactics to build support for military intervention in Central America in the 1980s).