What is special about the ratio $Q/V$ that we give it the name Capacitance? Why is the ratio Charge/Potential important? Also, usually when we add charge, the potential changes. Then why do we care how much charge we can put on a conductor for a given potential. 
 A: Here is an intuitive explanation:
Basically $C = Q/V$ means, if I have an capacitor such that store the same amount of charge $Q$ requiring a lower potential, then it has more capacitance. So, would be nice to build device which store huge amount of charge with ridiculous low voltage. This would require a huuuge capacitance. That's why $Q/V$ is important.
And yes, charge changes, potential changes. But the geometry can also change! For instance, plane capacitor capacitance $C_p$ and cylinder one $C_c$ are different:
$$
C_p = \frac{\epsilon_0 A}{d},\quad\quad C_c = \frac{2\pi\epsilon_0 l}{\ln b - \ln a}
$$
This assuming distance $d$ between the places, and inner and outer radius $a$ and $b$ from the cylinder one, and $l$ the length of the cylinder capacitor.
Also, assuming now plane capacitor. If you put a dielectric material between the plates, the new capacitance increases: $C = kC_p$, by a factor of $k > 1$.
A: As you can prove by using the uniqueness theorem, in an electrostatic system of charged conductors of whatever shape if the charge of each conductor is increased $n$ times, the electric potential at any point in space (with respect to the reference point) will also be increased $n$ times. Thus, the quantity $C=Q/V$ is  determined solely by the geometry of the problem (the shapes of the conductors and their relative positions).
A: Suppose we put charge on a conductor. The more charge we add to the conductor, the harder it gets to add more charge. That is, when we add charge, the potential at the conductor gets higher, and there will be a larger potential difference between the conductor and elsewhere; we will need a larger force to overcome that potential difference. 
We want to know how much harder we will have to work to add charge, i. e., how much will the potential change when we add charge. We consider the ratio Q/V; if V is high, we have to work hard to add charge - it's "harder" for the conductor to receive that charge. The ratio Q/V, then, is the ability of the conductor to hold charge; if V is small then we can add charge without applying much force, and so, the conductor has a high ability to contain charge. We call that ability capacitance
