A term often used for any amount of data that cannot be effectively or reasonably analysed or visualised with commodity computing hardware. The actual size of "big" varies depending on whether your commodity computing hardware is a single workstation or a high performance computing cluster. As of 2013, this tends to be roughly in the exabyte range, though BigData is also (and some might say, mainly) about processing infinite streams of data.
I've seen the term used to mean statistical "data mining" of large customer databases, logs, transactions, blog-entries, etc.
See also: SqlAndDataMiningDiscussion