What can DNA testing companies reveal about our ancestry and health, and at what risk?

Some 30 million Americans have sent their DNA to be analyzed by companies like 23andMe and AncestryDNA, hoping to obtain clues to family origins and forecasts of their future health. Some users have found family members and discovered lurking genetic risks. But what happens once the sample is in the hands of testing companies? What are they looking at and how accurate are their results?