Step 1 (Start from resistance ideas): Every conductor opposes current through its resistance \(R\); the material itself opposes current through its resistivity \(\rho\). Conductivity terms are just the opposites (reciprocals) of these two.
Step 2 (Electrical conductivity, i.e. conductance): It tells how good a given conductor is at carrying current and is defined as the reciprocal of resistance, \(G = 1/R\). A large \(G\) means current passes easily. Because current \(I = GV\), its unit siemens equals ampere per volt.
Step 3 (Specific conductivity): It tells how good the material is, independent of the sample's dimensions, and is defined as the reciprocal of resistivity, \(\sigma = 1/\rho\). In terms of microscopic quantities \(\sigma = \dfrac{ne^2\tau}{m}\), where \(n\) is free-electron density, \(\tau\) the relaxation time.
Step 4 (Units): \(G\) is in \(\text{ohm}^{-1}\) (siemens); \(\sigma\) is in \(\text{ohm}^{-1}\text{m}^{-1}\) (siemens per metre).
\[\boxed{G=\tfrac{1}{R},\qquad \sigma=\tfrac{1}{\rho}}\]