I suspect this heavy depends on the type of the aircraft, so lets assume we are considering a small airplane.
Absolutely right, it does vary widely. The parameter you're asking about is called glide ratio and it is directly related to another parameter called lift-to-drag ratio or L/D ratio. This is a fundamental characteristic of the aerodynamics of a particular aircraft. L/D ratio varies with airspeed; for determining best engine-out glide performance, the L/D ratio at "best glide airspeed" is used. "Best glide airspeed" is the speed that maximizes the L/D ratio, and this maximum value is known as L/Dmax.
The maximum L/D ratio (L/Dmax) of a Cessna 172 is about 9, so its glide ratio is about 9:1 - for every 9 units traveled forward it will lose 1 unit of altitude. So, it will glide about 9,000 feet for every 1,000 feet of altitude available. This is a fairly typical value for small planes.
To show you how widely variable this is, a modern glider can achieve ratios above 60:1, while the Space Shuttle ranged from about 1:1 at high speed, early in reentry, to 4.5:1 on final approach.
Notably, large transport aircraft tend to have significantly higher L/D ratios than small aircraft: a 747 can achieve an L/Dmax of about 17:1.
What governs the 'glideability' of the plane?
As above, its lift-to-drag ratio. Very casually speaking this is just a measure of how "aerodynamic" the airplane is, comparing its capability to generate lift with the drag it creates in the process. The better its lifting capabilities, or the less drag it generates, the higher the ratio.
For more information, I highly recommend a free online book on aerodynamics called See How It Flies by John Denker. It is written for pilots, not mathematicians or physicists, so it explains the concepts very intuitively without a lot of equations. It talks about L/D ratio and explains some of the factors that affect it. (I would recommend this book to any pilot anyway.)
Is it possible, for an airplane with an engine/engines, to leverage this to save fuel while flying, or are they too heavy/otherwise unable to do this?
Since glide ratio is directly related to (indeed it's the same thing as) L/D ratio, you could say that airplanes already do take advantage of it. The higher their L/D at cruise airspeed, the more fuel-efficient they will be (because less thrust will be required to counteract drag in steady-state flight).
If you're asking about shutting the engines off during approach/landing to save fuel, there is another question here on ASE which specifically addresses this. (The answer, in short, is no, it's not practical.)