Question: Just a few days ago, we were taught the basics of electricity in the IGCSE Physics syllabus. While he's said a great deal about potential difference and how it's the difference in voltage between two points in a circuit, he never bothered describing what "potential" itself was.

From what I have learned both from our physics teacher and the Internet, I guess it is actually just the short form of "potential electrical energy". 

If it is so, why is the term so less discussed?

Answer: 
It's a short form of *electrostatic potential difference*, which is the difference in the *electrostatic potential* between two points.

The eletrostatic potential is a field (a value associated with each point in space) that indicates how much energy would be required to put a charged particle at each point. But there is no energy associated with it until you also introduce that charged particle. 

Another way to think of this is that the electrostatic potential energy (of some configuration of charges) is the energy associated with the arrangement of those charges in an electrostatic potential. But the potential, and the potential energy are two separate things and you should be careful to understand how they're separate and how they're related.
Voltage (or potential difference) is somewhat related to energy in that the energy (W) in a capacitor is defined by: -

$$W = \dfrac{CV^2}{2}$$

And, if we use the charge equation Q = CV or V = Q/C we get this energy equation: -

$$W = \dfrac{Q^2}{2C}$$

If we then differentiated energy with respect to charge we get: -

$$\dfrac{dW}{dQ} = \dfrac{2Q}{2C} = \dfrac{Q}{C}$$

And this equals voltage. So the differential of energy with respect to charge is voltage.

Basically the voltage across (or between) two nodes is the ability to impart energy when moving charge. It's not very easy to understand really. It's a bit like the mechanical analogy that is more easier to understand: -

$$W = \dfrac{mV^2}{2}$$

This is the energy of a moving object (V= velocity) of mass m. But that mass also has momentum (P = mV). And, if we said V = P/m and plugged it into the energy equation of a moving body and differentiated with respect to momentum we would get: -

Velocity is the rate of change of energy with respect to momentum.

This is more easier to understand but is it meaningful to mechanical engineers. I don't think it is BUT is the voltage equation above meaningful to EEs. Yes it is but it's hard to grasp.

Current is child's play by comparison - electrons flowing past a point per second.

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Added section
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You can also start with power = voltage x current and, knowing that power is the rate of change of energy with respect to time: -

$$\dfrac{dW}{dt} = voltage \times current$$

But current is the rate of change of charge (the easy bit described above) it follows that: -

$$\dfrac{dW}{dt} = voltage \times \dfrac{dQ}{dt}$$

Or simply voltage = \$\dfrac{dW}{dQ}\$.