Damping coefficient physical meaning So I decided to calculate damping coefficient. I will make a mass on a string oscillate in a water (somehow) and then using camera will plot the graph of time versus altitude.
Now I know that the maximum altitude points together forms another line (exponential), which slope, when linearised, will give me the damping coefficient. My question is - where could I use this coefficient, what would be its practical value and is there anyway I could compare it with something just so it doesn't look completely random?
 A: Any formula or constant of Physics is practically used to predict (or explain/verify) a physical phenomenon. So your coefficient can be used for the same purpose, e.g. you do the experiment again, and you can predict the outcome from beforehand, in the limits of errors.
To get a simpler picture of damping, imagine a block attached to a string. The block is oscillating horizontally on a rough surface. The friction will damp the motion; try to get the picture. There is one difference though, here the damping force is mostly independent of velocity (kinetic friction), whereas the damping force in the water will depend on velocity. That's actually the reason that it appears more 'random'.
A: The damping coefficient for a vibrating system usually depends on many physical properties of the system and is very difficult to predict from first principles. 
In your experiment you could try to compare it with the drag force on a long cylinder moving through the fluid. If you measure the vibration frequency and amplitude, you can calculate the relative velocity of the string and the fluid - but the usual formulas for drag force assume the that flow is steady, which is not true for your vibrating string.
Often, a useful way to interpret the effect of damping is to look at the rate at which energy is being dissipated from the vibrating system. 
It might be interesting to do the experiment for different vibration frequencies (e.g. change the tension in the string) different diameters of the string, and fluids with different densities and viscosities (including air, as well as liquids). You might find there is a useful relationship between the damping coefficient and something analogous to Reynolds number - you will have to decide how to define an "average Reynolds number" for the non-steady-state motion.
A: It sounds like you are dealing with a second-order system based on the oscillatory nature you described. The governing equation for your system is simply,
$$ \frac{1}{\omega_n^2}\ddot{y} + \frac{2 \zeta }{\omega_n} \dot{y} + y = K F(t) $$
where $y$ is the displacement, $\omega_n$ is the natural frequency, $\zeta$ is the damping coefficient, $K$ is the sensitivity, and $F(t)$ is the forcing input function to your system, respectively. The value of $\zeta$ is going to govern the response of your system. For instance, here is a classic chart that demonstrates the response of a second-order system from a unit step input. 

Notice the varying degree of behavior based on the different damping coefficients $\zeta$ for each system. If $\zeta = 0$, the system is undamped and the step input will simply send the system into undamped free oscillation. If $0 < \zeta < 1$ we call the system underdamped and we observe prolonged oscillatory motion as the system returns to the value of the step input. If $\zeta = 1$ the system is critically damped, and there are no observed oscilations from a step input. Lastly, if $\zeta > 1$ then the system is overdamped. From a step input, you can easily determine the systems damping coefficient as you mentioned. It is termed the logarithmic decrement method and is given by,
$$ \zeta = \frac{\ln (y_1/y_2)}{\sqrt[]{4\pi^2 + \ln\left(y_1/y_2\right)^2}} $$
It is not entirely clear what your goal is with this setup. Are you setting up a pendulum system in water? That was my take away, but I may have misunderstood your description. If so, then I would anticipate a damping coefficient $\zeta$ to be less than 1, but probably on the high end near unity. Say $\zeta \approx 0.7-0.9$. Water is a fairly viscous fluid and the viscous dissipation will be large, but probably not enough to critically damp or even overdamp the system. 
If you are considering oscillating the mass up and down in the water, then you will most likely be providing some form of a sinusoidal input. In this case the response of your system for different values of $\zeta$ will look similar to these classic forced vibration charts.

The main takeaway for you is how $\zeta$ changes important features of the behavioral response like the magnitude ratio (amplitude response of your system) and the phase shift. However, this is all dependent on what is your goal of your system. 
