Routing trace characterstic impedance determination based on interface
The impedance is determined by the dielectric (air/FR4/teflon/rogers/etc.) and the geometry of the trace(s) and surrounding ground plane(s).  There are calculators available online and in various software packages that can calculate various parameters.  Generally you will input the copper thickness, ground plane spacing (board or layer thickness), dielectic constant, desired impedance, etc. and it will tell you the required trace width.  

If you connect a 50 ohm driver to a 20 ohm trace you can run in to problems with reflections and intersymbol interference.