The point of this question is to compile a list of applications of hyperbola because a lot of people are unknown to it and asks it frequently.
Dulles Airport, designed by Eero Saarinen, has a roof in the shape of a hyperbolic paraboloid. The hyperbolic paraboloid is a three-dimensional surface that is a hyperbola in one cross-section, and a parabola in another cross section.
This is a Gear Transmission. Greatest application of a pair of hyperbola gears:
And hyperbolic structures are used in Cooling Towers of Nuclear Reactors.. Doesn't it make hyperbola, a great deal on earth? :)
Did you ever take a look at the light projected onto a wall by a nearby lamp with a standard lampshade? That's right: the light on the wall due to the lamp has a hyperbola for a bounday. The reason for this is clear once you think about it for a second: the light out of the lampshade forms a vertical cone, and the intersection of a vertical cone and a vertical wall makes a hyperbola.
Also, consider a pair of sources of ripples in water that produce concentric waves. The intersections of those concentric waves - surfaces of constant phase, are hyperbolae. Why? Because a hyperbola is the locus of points having a constant distance difference from two points (i.e., a phase difference is is constant on the hyperbola).
Hyperbolas are used extensively in economics and finance (specifically portfolio theory), where they can represent the various combinations of securities, funds, etc. that yield similar risk-return ratios. This is why you often see efficient portfolio frontiers represented as partial hyperbolas. For similar reasons, production frontiers, which represent various combinations of capital and labor that produce a given output, as hyperbolas.
I don't know if that's entirely a "real-world" example because it's not a tangible object, but the mathematics of hyperbolas are still very important.
For example, the upper edge of this hyperbola (the part of the curve above the inflection point) in this plot:
represents the optimal combination of two risky assets, assuming the portfolio doesn't contain any risk free assets like Treasury bills. In this case, an optimal allocation is one that provides the highest ratio of expected return to risk, i.e. standard deviation. This is also known as the Sharpe Ratio.
In addition to the awesome answers, here is something mundane: a hyperbola occurs whenever you have a formula of the form $$xy = c$$ Two hyperbolas, if you consider negative values. Equations of this form crop up all over the place, in natural sciences, economics, you name it.
Then, in space, when a small mass passes by a large one (say, comet around a planet), and it is moving faster then escape velocity with respect to the large one, its path is hyperbolic.
Most receptors are made in the shape. e.g. RADARs, television reception dishes, etc. because they need to reflect off the signal and focus it on a single "point".
When two stones are thrown in a pool of water, the concentric circles of ripples intersect in hyperbolas. This property of the hyperbola is used in radar tracking stations: an object is located by sending out sound waves from two point sources: the concentric circles of these sound waves intersect in hyperbolas.
In computer science, it's the shape of the response-time curve for request-reply pairs. It starts off parallel to the x-axis at low loads, curves upwards and ends up approaching parallel to the line y = (Dmax * x) - Z, where Dmax is the service demand of the slowest part of the system and Z is the user think time between requests.
Hyperbolas are used extensively in Time Difference of Arrival (TDoA) analysis, which has many applications. For example, it is used for geolocation to determine the location of a vehicle relative to several radar emitters (e.g. passive geolocation of UAVs), localizing cellular phones without requiring a GPS fix (e.g. U-TDOA), or making "tapscreens" that can sense the precise location of a tap on a large display without expensive touchscreens (e.g. MIT's Tapper).
In TDoA, multiple sensors each detect the arrival time of a particular signal. The time differences between any two sensor measurements define a hyperbola of possible origin locations (since those are the points with a constant difference in distance to each sensor). Intersecting the hyperbolas gives you the position of the signal's source very quickly and precisely.