What does electrical potential at a point mean? From my understanding, potential difference (or voltage) between point A and point B is the difference in electrical potential at the two points. The potential difference is also, the work done per unit charge in moving charges from point A to point B.
But apparently, the potential at one point (e.g point A) is also measured in voltage? How is it possible that at point A, there is work done per unit charge in moving charges from point A to point A?
 A: You can think of it in such a way. Let $\phi(x)$ be the potential function of an electric field $\vec{E}$ with boundary conditions $\lim\limits_{x\to \infty} \phi(x) = 0$. For the potential we obtain
$$\phi(x) = \phi(x) -\phi(\infty) = \int_\infty^x \quad \vec{E}(y) dy.$$
If you bring your charged object from the point $x$ to spatial infinity you obtain the difference $\phi(x) -\phi(\infty) = \phi(x) - 0 = \phi(x)$. 
It's a matter of choosing the zero level of the potential. 
A: In the definition of electric potential at a point in space, the reference point is usually assumed to be at infinity. So, the potential at point A is defined as the work that needs to be performed to move a positive unit charge from infinity to point A.
A: You have understood the concept of potential difference between two points A and B. But you are confused as to why there is a potential at a single point, say P. When we talk about the potential at a point P, we usually refer to the potential difference between infinity and the point P. That is, the potential at point P is defined as the work done in bringing a unit positive charge from infinity to that point. Its simple, just replace point B in your given example with infinity and point A with point P. 
A: In common introductory textbooks, voltage is defined as the electric potential difference (say, between A and B) between two points in space. They say that A and B have a potential difference, and both A and B are at unique electric potentials.
This is completely narrow viewpoint.
Electric potential, electric potential difference, and voltage are all completely synonymous. 
In your case, you are right that it doesn't make sense to define a unique potential at point A. When you say the "potential at point A", what you really mean is the difference in electric potential between point A, and some explicit (e.g., circuital ground) or even implied (e.g., infinity) point of reference.
After all, no potential is absolute. When you say a ball "has potential energy $mgh$", you are secretly saying that "the gravitational potential energy difference between its height and the ground is $mgh$". You could also say that its gravitational potential is zero at that height, in which case it would have potential $-mgh$ at the ground. 
It's all relative to one's choice of reference point -- and this choice is always made, either explicitly or implicitly.
