What's the significance of a Killing horizon? A Killing horizon is defined as a null hypersurface generated by a Killing vector, which is then null at that surface. Some often cited examples come from the Kerr spacetime, where the Killing vector $\partial_t$ becomes null at the ergosphere; one can also take a linear combination of that with $\partial_\phi$, which is null at the event horizon.
From this it would seem that Killing horizons are related to "special" surfaces of the spacetime, although not always: Minkowski spacetime has Killing horizons at any point, due to its high degree of symmetry. Can we make some general statements about Killing horizons besides what the definition says? Are they related to event horizons in any predictable way? If I have a metric and I find a Killing horizon, what can I say about it?
 A: The "special" surfaces of spacetime defined by Killing horizons are null hypersurfaces. A null hypersurface which is a hypersurface whose normal vector at every point is a null vector (with respect to the local metric tensor).
The "boring" and trivial example is a light cone, as already mentioned. EDIT: from the comments, it is true that this statement is not true for Minkowski spacetime. Then I must say I am not sure when this applies.
In terms of other applications, I can think of two, even though they are quite interconnected: black holes event horizons, and surface gravity $\kappa$. A nice set of slides with useful discussions about this can be found here: 
Black holes
Taken verbatim from here:

[...] by Hawking’s rigidity theorem the event horizon  of  a 
  stationary,  asymptotically  flat  black  hole  spacetime 
  (supplemented  by  certain  additional assumptions, see [19] for a
  review), is a Killing horizon. In fact, one often uses the notion of a
  Killing horizon to provide a quasi-local definition of an equilibrium
  black hole.

I just wanted to show the above as a "quantitative" connection to the event horizon, as you asked. In this case, then, you can see that a physical meaning (albeit asympotically) of a Killing horizon is that it corresponds to the event horizon.
But black holes are also entering the picture through surface gravity. See below.
Surface gravity
Surface gravity has a meaning in Newtonian/classical gravity, which is not the same in GR. Maybe the same name was used, historically, because one wanted to define the same object. But the two things have differences nowadays. Especially in black holes.
The physical meaning of the GR surface gravity $\kappa$ of a static Killing horizon is the acceleration, as exerted at infinity, needed to keep an object at the horizon. Mathematically, if $k^{a}$ is a suitably normalised Killing vector, then the surface gravity is defined by 
$$   k^a ∇_a k^b = \kappa k^b , $$
where the equation is evaluated at the horizon. Specific solutions for black hole metrics are listed here.
Surface gravity is "physically" interesting because it is related to the temperature of Hawking radiation $T_{\mathrm{H}}$:
$$T_{\mathrm{H}} = \frac{\hbar c\kappa}{2 \pi k_{\mathrm{B}}}.$$
