The Aharonov-Bohm effect is a variation on the double slit experiment using charged particles (usually electrons) with some unique consequences: it provides direct evidence that the potential is more 'real' than the field (assuming certain things about reality, of course).
The only change to the setup is that a magnetic field is established and varied. This magnetic field threads in between the paths of the electrons. The electrons do not pass through the field; rather, the two paths of the electrons enclose the magnetic field.
"Well, then, there's no force, nothing changes, right?" one can reasonably ask. If you block off one slit, thus bypassing the interference step, then yes; nothing changes. This is important: it establishes that in fact there is no force applied by the magnetic field.
However, there is a change. Rather than viewing the situation as there being a magnetic field B, we can look at it as the corresponding vector potential, A. If the magnetic field was oriented straight through the gap, then the vector potential is curled around it (and through it; the important part is, it curls around). So, the vector potential DOES extend into the region through which the electrons pass.
For all wavefunctions, the rate of change of phase is equal to the hamiltonian applied to the state (a restatement of Schrodinger's equation). The hamiltonian of a charged particle in a magnetic field includes the term (p - A)², with p the momentum. So, if A and p are in the same direction, the hamiltonian will be smaller, and if A and p are in the opposite direction, the hamiltonian will be bigger.
Here we get to the juicy part: since A wraps around the central field, then of the two paths the electron takes, one of them points along A, while the other points against A. So, the two electrons will see a different phase shift. This phase shift produces a change in the interference pattern.
If an experimenter varies the magnetic flux, different phase shifts can be observed. Once the magnetic flux gets to be strong enough, the phase shift is one complete cycle, and the interference pattern is exactly where it started. This amount of magnetic flux is sometimes known as an Aharonov-Bohm cycle.
This gives us a reason to 'prefer' the vector potential over the magnetic field: If we consider the magnetic field 'real' and the vector potential to be a human invention, then we must allow the laws of physics to be nonlocal -- where an electron ends up must depend on things that are not where it is. This is generally considered bad form for a physical theory. If on the other hand we consider the vector potential to be real and the magnetic field to be a human invention, everything remains local.
On the other hand, the vector potential, like any other potential, has the disadvantage that it cannot be directly measured or determined -- we can only measure the magnetic field, whether directly, or indirectly via this effect... and that is not enough to fully determine the vector potential. You can always add a curl-free field to the vector potential and the magnetic field will come out exactly the same (this is known as Gauge invariance). If this underdeterminedness bothers you, well, both you and it aren't alone: some scientists agree, and there are a variety of other underdetermined parameters in nature. Other underdetermined parameters are our absolute position in space-time, the background electrical potential, the background gravitational potential, absolute quantum phase, etc. Some time in the future, laws which break this symmetry may be discovered*. Until then, we can stick to the measurable OR keep locality.
*General relativity does provide a way of extracting the background potentials, but only in the limit that energy makes sense. It doesn't always need to in GR, which makes one wonder what is really going on.
Also, unperson says: ... I'd be a bit careful in making statements about implications on locality. If you simply mean that our description of nature is non-local, then this is true of QM either way, as is manifest in an entangled state. If you're talking about issues of causality, either description in terms of fields or potentials should have them same implications for transmission of information.
The specific non-locality in question here is nothing so subtle as entanglement. Put directly, nowhere in quantum mechanics does the way that the time evolution operator affect a particular event depend on anything a macroscopic distance away, or at a time any earlier than the previous moment. It only depends on the past time-like neighborhood of that event. This is the very heart of locality. Things only ever happen because of stuff that's right at them.
If we only accept the reality of the field and not the potential, this is no longer the case - the time evolution operator must look afar to find a magnetic field, or into the past to find an electric field, so it can properly set the phase. It is this that is objectionable.