Most calculators will display only a certain number of digits. The calculator I normally use (a TI-84) only displays 10 digits, so when I ask it what pi is, it spits out the familiar 3 and nine decimal places:
Calculators store more digits in their memory than they can display however, and through 'shifty' 'trickery' that only Mathmagicians know, you can squeeze out a few more digits. I always feel like I'm torturing the calculator when I do this, like taking years away from Wesley's life in Princess bride, but here goes anyway.
Simply subtract the 3, and recalculate: I get .1415926536
The first thing I noticed is that I got most of the same digits, except the 4 has become a 3, and now a 6 shows up. That 6 (or maybe its a 5... we'll see) is the 10th digit of pi, and because it was 5 or above, the ninth digit 3 had to be rounded up to a 4 in the first example.
1.415926536 - 1 becomes .4159265359 and I see that the 10th digit of pi is actually a 5, not six, and the 11th digit is a 9 (or is it an 8?).
This process can be repeated over and over again to squeeze more digits out, but only to a point. For the TI-84, that is only 13 digits long, yielding a stored value of pi which is:
3.1415926535898 or 3.1415926535897
As shown above, we can't know for sure if the 13th digit has been rounded or not because we don't know what the 14th digit is, so it's either a 7 or an 8. Wolfram Alpha to the rescue reveals that it is indeed a 7, and lists many more digits for you, if you'd like.
How many digits are there? As of October 17, 2011, over 10 trillion were known, which is enough to give everyone on earth a digit, and still have plenty left over. Computers are constantly working at finding more digits, using even shiftier trickery than this. What's really creepy, is that your phone number shows up in pi! And so is mine, and my neighbors. It must be stalking us!