To me (coming from a relational database background), "Big Data" is not primarily about the data size (which is the bulk of what the other answers are so far).
"Big Data" and "Bad Data" are closely related. Relational Databases require 'pristine data'. If the data is in the database, it is accurate, clean, and 100% reliable. Relational Databases require "Great Data" and a huge amount of time, money, and accountability is put on to making sure the data is well prepared before loading it in to the database. If the data is in the database, it is 'gospel', and it defines the system understanding of reality.
"Big Data" tackles this problem from the other direction. The data is poorly defined, much of it may be inaccurate, and much of it may in fact be missing. The structure and layout of the data is linear as opposed to relational.
Big Data has to have enough volume so that the amount of bad data, or missing data becomes statistically insignificant. When the errors in your data are common enough to cancel each other out, and when the missing data is proportionally small enough to be negligible. When and when your data access requirements and algorithms are functional even with incomplete and inaccurate data, then you have "Big Data".
"Big Data" is not really about the volume, it is about the characteristics of the data.
