We usually use an infinite series that converges to pi or some other iterative method.
For example arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ... up to infinity. If x=1 then arctan(1) = pi/4 so you can see that you could calculate pi in such a way. The problem is that this infinite series converges (approaches) pi very slowly. What this means is that you need to add a lot of terms before the number starts to look like pi.
Currently, there are much better infinite series to use for example: http://planetmath.org/encyclopedia/RamanujansFormulaForPi.html [Broken] is a much better infinite series to use to calculate pi.
Typically the computations are done, well, by computers that are able of doing calculations very fast. I think that currently pi has been calculated to like 10 billion digits or something ridiculous like that.