Big data is basically just statistical analysis done on a really large scale (data sets as big as "All Google searches performed on March 12th, 2012, etc.) in order to extract useful information about the people who use services that leave a digital footprint to read.
"Big data" isn't really that new a concept, but it's just become more visible in public awareness due at least partially, in my opinion, to the growing affinity that your average media-consumer has for technology. Movies like Moneyball and A Beautiful Mind that highlight the superiority of analysis over "gut" in terms of making business decisions have led to increased media coverage of similar topics.
With regard to how it effects the regular consumer - big data isn't something to be afraid of. The whole reason why big data works is because it takes information on a scale large enough to minimize deviations caused by individuals on their own. So, concerns about how big data is a privacy threat or something like that are exaggerated. In other words, big data concerns are nothing different or new than the same old privacy concerns that people have always had about companies gathering information through their digital footprint.