Converting a simple analog Butterworth filter into a digital filter
Converting the analog filter $H_a(s)$ into a digital filter $H_d(z)$ using the bilinear transform where T is the sampling period:

$\Large H_d(z) = H_a(s)\bigg|_{s=\frac{2}{T}\frac{z-1}{z+1}}$

Example:

Given a first order Butterworth filter

$H_a(s) = \frac{1}{1+RCs}$

$H_d(z) = H_a\bigg( \frac{2}{T} \frac{z-1}{z+1} \bigg)$

$H_d(z) = \frac{1}{1+RC\Big(\frac{2}{T}\frac{z-1}{z+1} \Big)}$

$H_d(z) = \frac{1}{1+RC(\frac{2}{T}\frac{z-1}{z+1})}$

$H_d(z) = \frac{1+z}{(1-2RC/T)+(1+2RC/T)z}$

$H_d(z) = \frac{1+z^{-1}}{(1+2RC/T)+(1-RC/T)z^{-1}}$

The coefficients of the denominator are the 'feed-backward' coefficients and the coefficients of the numerator are the 'feed-forward'  coefficients used to implement a real-time digital filter.