What is gamma in the damping equation? $x''+\gamma x'+w_0^2x=0$ That is the general equation for damped harmonic motion. What is the term or name that describes gamma ? Is it called the damping constant ? I know its the ration between the resistive coefficient (b) and mass of the system (m) but what do we actually call it ?

The second order differential equation arises from the application of Newton's Second Law.

$$\sum F = ma$$ In the case of oscillatory systems, such as a spring, there are two forces exerted onto the spring. The restoring force $-kx$ and the damping force $-bv$ where $v$ is the velocity of the spring. Note that the assumption of linear resistive force is only an approximation, and at higher velocities drag is actually proportional to the square of velocity.

$$\begin{align} -kx-bv &= ma\\\ m\ddot{x} + b\dot{x}+kx&=0\\\ \ddot{x}+\frac{b}{m}\dot{x}+\frac{k}{m}x &= 0 \end{align}$$ which is the equation you had above.