How do you know one approximation to $\pi$ is more accurate than another?

We can prove that $$\pi=4\arctan 1 =4\sum_{i=1}^\infty \frac{-(-1)^i}{2i-1}=4\left(1-\frac 13+\frac 15-\frac 17+\dots\right)$$ There are series that converge much more quickly, but let's pretend that this is the only series so far known. The alternating series theorem says that the error truncating a series like this is of the sign of and smaller than the first neglected term. A lazy mathematician might compute the first hundred terms, getting $3.13159\frac {+0.02}{-0}$. Another might sum the first thousand terms, which would give a maximum error of $0.002$ and would clearly be better. Other formulas also come with a bound on the error, so we know how bad it can be.

An easy to understand method of approximating pi is that of polygonal approximation. As is the case here, we often can't quite state the exact value of pi. However, we can say with certainty that pi lies between some lower bound and upper bound.

If the lower and upper bound are within $10^{-n}$ of each other, then we can state with complete certainty the first $n$ digits of pi, since these digits are shared by the upper and lower bound. The challenge, then, is to get the upper and lower bound as close together (and thus as close to pi) as possible.

One of the most common ways to calculate $\pi$ is through a sum of a series.

A simple one to see is the Taylor Series of $\arctan(x)$ evaluated at $1$ yields $\pi / 4$. (Link for WolframAlpha on this Taylor Series: http://goo.gl/XK5O1i). Its easy to see that the sum of the first $n$ terms of the Taylor series is as an approximation of $\pi$ that is less accurate than the sum of the first $n+1$ terms.

This guarantees that we can continuously improve the quality of our approximation of $\pi$. Now the method I just showed is not the fasted way to approximate $\pi$ (by this I mean the one whose error after $n$ steps is the least), but it is certainly one way that we can guarantee continual improvement of our approximation by increasing the number of terms we are using.

Hopefully this helps!