OK, I will assume you have the under-damped case.
If you continue reading the wikipedia article in question you'll find the solution for a underdamped oscillator writen as
$$ x(t) = e^{- \zeta \omega_0 t} (A \cos(\omega_\mathrm{d}\,t) + B \sin(\omega_\mathrm{d}\,t )) $$
with $A$ and $B$ constant.
So, take you data, and plot all the maxima (or minima) as a function of time, fit an exponential {*} to that and $\zeta \omega_0$ pops right out.
If you also need to get $\omega_o$ from the data use
$$ \omega_\mathrm{d} = \omega_0 \sqrt{1 - \zeta^2 } $$
where you get $\omega_\mathrm{d}$ by extracting the average period (i.e. time from peak to peak) in the data and noting that the period is $T_d = \frac{2 \pi}{\omega_\mathrm{d}}$.
Now you have two equations for two unknowns, so all you have left is a bit of algebra.
{*} Or plot amplitude versus time on semi-log paper if you are doing this the old-school way. Or plot log(amplitude) versus time on linear--linear graph paper. Then extract the slope.
Assuming your car can be accurately described by a dambed harmonic oscillator when oscillating undisturbed, the differential equation defining its vertical motion is:
$$
\frac{d^2y}{dt^2}+a\frac{dy}{dt}+by=0
$$
Where $a$ is the damping coefficient and $b$ is the spring constant. It's characteristic equation is:
$$
r^2+ar+b=0
$$
In principal, there are three different cases depending on the nature of the roots of this equation, but assuming your car will exhibitit exponentially decreasing oscillations, we can assume that it has two complex roots, $r_1=\alpha+i\beta$ and $r_2=\alpha-i\beta$ and the general solution to the differential equation will be:
$$
y=e^{\alpha t}\left(C_1\cos\beta t+C_2\sin\beta t\right)
$$
If you initialize oscillations with large amplitude in your car manually and then record $y(t)$ with a camera, you should be able to estimate $\alpha$ (basically by registering how fast the amplitude decrease between each oscillation). Note that $\alpha$ should be negative. The easiest way to do this might be to plot the natural logarithm of the highest point reached for each oscillation against time. You should get a straight line with $\alpha$ as slope. Just remember using the car's rest position as baseline when measuring $y$.
Once you have $\alpha$, just use that:
$$
(r-\alpha-i\beta)(r-\alpha+i\beta)=r^2+ar+b\\
\Rightarrow r^2-2\alpha r+\alpha^2+\beta^2=r^2+ar+b\\
\Rightarrow a=-2\alpha
$$
to calculate your damping coefficient.
Best Answer
You should already be familiar with damping. It simply refers to the fact that if you set a spring going, it eventually stops.
The wikipedia article should cover most of what you want to know.
Any particular spring may be damped for all sorts of reasons. Any way the spring can lose energy contributes to damping, so it could be lost internally to heat due to stressing the material, or externally to heat via friction, or externally to an electromagnetic field, to some sort of mechanical dashpot, etc.
If you were designing an experiment to study damping, you could be interested in a number of different particular things. You will have to make your own choice about what the most interesting thing to study is. For example, would you like measure the damping ratio? The overall magnitude of the damping effect? The time it takes your spring to reduce its amplitude to 1/2 its previous value?
If you have a particular effect picked out, do you want to know how it depends on the load on the spring, or the material of the spring, or whether the spring is in a vacuum, etc. There are many possible things to study, so your question has no definite answer. I'm sure you can find something particular you find worthwhile.