How Butterworth low-pass filter can be applied on a digital signal (i.e. discrete representation of a signal)?
What you need to do is find a digital filter $H(z)$ that behaves similar to the $H(s)$ of your Butterworth filter. That is, you need to read about IIR implementation.

There are several methods described in signal processing books about this subject. I recommend you to check for example "Essentials of Digital Signal Processing" from Lathi and Green. 

Two of the most common methods are the "Impulse invariance" and the "Bilinear transform".


The impulse invariance requires you to find the impulse response $h(t)$ of your $H(s)$. After doing this, you sample it replacing $t$ by $nT$, to finally Z-transform your $h[n]$ into $H(z)$. Bare in mind that this method only works for band-limited filters (lowpass and bandpass) because of aliasing issues. You should determine an appropriate small enough T to avoid this.
The bilinear transform, on the other hand, consists on applying the following mapping of your variable $s$ into $z$:   

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

This method has a drawback as well, which is the frequency warping, you can read about it here https://en.wikipedia.org/wiki/Bilinear_transform#Frequency_warping, but is in general easy to counter by pre-warping your original $H(s)$.