For our statistical definition of entropy, we use the equation s = klnw. in this equation, what does k represent?

In the natural logarithm format or in equivalent notation (see: logarithm) as:

base e assumed, is called the Planck entropy, Boltzmann entropy, Boltzmann entropy formula, or Boltzmann-Planck entropy formula, a statistical mechanics,  S is the entropy of an ideal gas system, k is the Boltzmann constant (ideal gas constant R divided by Avogadro's number N), and W, from the German Wahrscheinlichkeit (var-SHINE-leash-kite), meaning probability, often referred to as multiplicity (in English), is the number of “states” (often modeled as quantum states), or "complexions", the particles or entities of the system can be found in according to the various energies with which they may each be assigned; wherein the particles of the system are assumed to have uncorrelated velocities and thus abide by the Boltzmann chaos assumption. 

I hope this helps.