What is a potential well? What exactly is a potential well physically?  [I've read the linked Wiki article, but it doesn't answer my questions, such as, what does it mean for a particle to "move along a potential" and "roll down potential"? How does this correspond to the particles actual motion in space?] 
I've read many examples where the author talks of a particle being trapped in a potential well and requiring a certain amount of energy to "climb" out of it, but how does this relate to the particle's actual motion, and how does it become "captured" in a potential well?
Heuristically, is it simply the case that a potential well is present in a particular region of space due to a action of a force in that region of space. The force acts on particles within that region and so this requires a given particle to do work against the force if it is to carry on along its original trajectory. If the particle has less energy available to it than the energy required to overcome the force acting on it (i.e. its kinetic energy is less than the potential energy associated with the force) then the particle will be "trapped in the potential well", i.e. confined to travel within the region in which the force is acting along trajectories prescribed by the action of the force. If the particles kinetic energy is greater than the potential energy (associated with the force), then it won't be "trapped in the potential well" and will be able to "escape" the region of space in which the force is acting.
As an example, in simple harmonic motion (SHM), a spring is subject to a restoring force $F=-kx$ and this has an associated potential $V=\frac{1}{2}kx^{2}$. The potential well is then a quadratic, with the minimum describing the equilibrium configuration of the spring. The spring then oscillates about the minimum of the potential - compressing as it moves up one side of the potential (hence gaining potential energy and losing kinetic energy), then moving back to the minimum of the potential well (losing potential energy and gaining potential energy) and then stretching as it moves up the other side of the potential (again gaining potential energy and losing kinetic energy). It then repeats this in an oscillatory fashion.
 A: A potential well is basically the energy required to be in a certain position. For example, if you have a hole, and a ball in the hole, it will require a certain energy for it to get up and out of the hole. However, if it does not have that energy, it's stuck at the bottom of a hole. Here's an illustration of this:

This is a basic example, but another could be something like a gravity well, as in General Relativity. Just think of spacetime as having a dip in it...it requires whatever object, the Moon, say, a certain amount of energy to get out of that dip and escape the Earth's orbit. 
Now, quantum mechanics makes this all a little more interesting. Just like before, if you have, say, a particle, it requrires a certain energy to get out of the potential well. Believe it or not, this occurs in the atom, because of the strong nuclear force. Within the nucleus, it is practically impossible for the protons and neutrons to get out because of the strong nuclear force, but once they get just out of range, they can leave very quickly. A diagram of this is shown below:

Now, you might ask, what in the universe is that little arrow that goes through the potential well? Isn't the whole point that the particle can't just get through that? Well, actually, in the weird and whacky world of the quantum, it can, and it's called quantum tunneling. The particle has a chance of just kind of sneaking through the barrier, and popping out on the other side before the universe notices anything wrong. Quantum tunneling makes nuclear fusion possible, and is one of the phenomenon that makes people look at QM a little funny.
Hope this helps!
A: Go to the top of a hill and dig a hole and but a ball in it. The ball must rise to exit the hole, before it can fall freely all the way down.
This is a potential well. A barrier. Your spring example is correct; though with the comment that the spring will never exit that potential well (there is nowhere to exit to - there is no maximum potential energy - it is not a "hole", which the string can get out from).
