Today is Pi Day because if you write the date in decimal format, you get 3.14 - which is the first three digits of the number pi: 3.1415926.... Pi is an irrational number, which means that (as far as we know) the digits will never repeat or terminate - the decimal expansion will just keep going on and on and on forever. That's why we have the handy-dandy Greek letter π to represent it, so that we can actually calculate with it; 3.14 is the most commonly-used approximation of pi now, but scientists use more decimals and many cultures have used 3 as an acceptable approximation.
There are mulitple ways to calculate pi, including a method using Fibonacci numbers. Calculating pi is a common test of raw computing skill, which may be why over 1 billion digits have been calculated so far. But there also seems to be something more to it, some "because it is there" challenge that keeps people interested in it; maybe it is the quest to see if there really is no pattern to the digits. Seems like mindless entertainment to me, but I suppose someone has to find it fun.