The fundamental law of extinction states that the extinction process is linear in the intensity of radiation and amount of radiatively active matter, provided that the physical state is held constant. (Neither concentration or length are fundamental parameters.)  There are two factors that determine the degree to which a medium containing particles will attenuate a light beam:  the number of particles encountered by the light beam, and the degree to which each particle extinguishes the light.  
For the case of absorption (Beer), this later quantity is called the absorptivity [], which is defined as "the property of a body that determines the fraction of incident radiation absorbed by the body".  The Beer-Lambert law  uses concentration and length in order to determine the number of particles the beam encounters.  If we know the area of a collimated beam (directed radiation), we can get the number of particles in a distance. The number of particles encountered can be calculated from Avogadro's number, the molar concentration, the cross-sectional area of the incident beam .   
There must be a large number of particles that are uniformly distributed for this relationship to hold. In practice, the beam area is thought of as a constant, and since the fraction [] has the area in both the numerator and denominator, the beam area cancels in the calculation of the absorbance.  The units of the absorptivity must match the units in which the sample is described.  For example, if the sample is described by mass concentration (g/L) and length (cm), then the units on the absorptivity would be [ L g cm], so that the absorbance has no units. 
For the case of "extinction" (Bouguer), the sum of absorption and scatter, the terms absorption, scattering, and extinction cross-sections are often used.  The fraction of light extinguished by the sample may be described by the extinction cross section (fraction extinguished per particle). the number of particles in a unit distance and the distance in those units.  For example:  [ (fraction extinguished / particle) (# particles / meter) (# meters / sample) = fraction extinguished / sample ]