Ballistic efficiency is the ratio of the kinetic energy of the projectile to the total energy released by the propellant. The kinetic energy of the projectile (at the muzzle) is 0.5mv^2. (Efficiency is about comparisons, so you can ignore the absolute size of the numbers and just think about comparing them across loads.) The energy released by the propellant is the product of the volume of the propellant and the volumetric energy density of the powder. QuickLOAD shows you powder energy density, which usually falls in the range 3-4 kilo-Joules per cubic centimeter. QuickLOAD also displays projectile energy in Joules (which, again, varies with bullet weight and muzzle velocity), and ballistic efficiency as a percentage, in the lower-right corner.
For example, a 280 AI with a 27" barrel shooting a 180-grain Berger VLD-H, starting with a COAL of 3.579", at 70 degrees F, in front of 57.6 grains of IMR 4831, achieves 31.8% ballistic efficiency. That's a pretty good number, even though the muzzle velocity is a modest 2,855 fps. IMR 4831 only has an energy density of 3.2 kJ/cm^3. Switch to 64.6 grains of RL-26 (energy density 3.9 kJ/cm^3) and you'll get a muzzle velocity of 2,995 fps at an efficiency of 29.7%. Less efficiency, more speed. Which do you care about?
Many factors can influence ballistic efficiency. Choice of cartridge, barrel length, brass (with a given internal volume), bullet weight (which resists acceleration, and other bullet characteristics, e.g. the surface area of the bullet shank, which relates to barrel friction, and the cross-sectional area of the bullet), powder (which has properties such as energy density and burn rate)--these and more can affect how well the powder pushes the bullet in a given gun on a given day. For example, a powder that burns too slowly for the barrel length, cartridge, bullet, etc. won't burn completely (QuickLOAD: "Amount of Propellant Burnt," bottom left corner of the lower-right window), and so won't transfer as much energy to the bullet before the bullet escapes the barrel. That would be an "inefficient" powder in that situation.
The hypothesis that a "short, fat" cartridge is in some sense inherently more efficient than a "long, thin" cartridge depends in part on assumptions about powder energy density and burn rate. Spend some time experimenting with QuickLOAD or Gordon's Reloading Tool, learning to match powders to bullets, cartridges, and gun characteristics such as barrel length, and you'll see that many variables affect ballistic efficiency--that it's by no means simply a property of a given cartridge geometry. Traditional "long, thin" cartridges such as the 280 AI can achieve pretty high efficiencies with the right combination of bullet, powder, and barrel length. My impression from years of doing this sort of analysis (using QuickLOAD) is that the ratio of brass length to brass diameter has a modest effect on ballistic efficiency (and is mostly overshadowed by other practical considerations). My theory about why this might be so has to do with the ability to burn the entire (relatively short) powder column well before the bullet escapes the barrel, so all of the powder has converted to hot gas sooner and so pushes (as much as possible, because we've created as much hot gas as possible behind the bullet) longer. That may mean we can reduce a barrel length from 26" to 24" to achieve a full burn at a given muzzle velocity.
The tradeoff--if that theory is correct--is that you get higher pressure peaks by using faster-burning powders. Higher peak pressures burn throats faster and may make it harder to achieve low extreme spreads in muzzle velocity. Also, less total powder means a short/fat case can't push the heaviest bullets of a given diameter as fast as cases having higher volume (long/fat cases like the 7mm Rem. Mag., the 300 RUM, or the notoriously inefficient 30-378 Weatherby, whose long free bore is designed to let hot gases blow around the bullet at the moment of ignition, to avoid excess peak pressures while leaving a huge amount of powder to propel the bullet after that first moment). At some point the only way to push the heaviest-for-caliber bullets is to put a whole bunch of powder behind them.
I have come to prefer cartridges that like very slow-burning powders, which can continue to burn more steadily and hence increase the amount of gas at a more nearly uniform rate, thereby flattening out the pressure curve (QuickLOAD: red bell curve, lower left), as the volume inside the barrel (behind the bullet) increases with the bullet's forward travel. In my experience this approach achieves high muzzle velocities with low extreme spreads, often keeping peak pressures under or near 60kpsi, where fast throat erosion becomes pronounced. (Cartridges such as the 6mm CM have been designed precisely to allow the use of very slow-burning powders for a given bullet diameter, to achieve high muzzle velocities while extending barrel life by avoiding high peak pressures. For me this has become an important consideration. Barrels are expensive, and sometimes a barrel swap means doing without a rifle for months.) This formula consistently lets me find sub-half-MOA loads just shooting a 10-shot ladder.
For example, using slow-burning powders that I selected for high energy content and high fill ratios, and shooting long-for-caliber high-BC projectiles, I easily exceed 3,000 fps with terrific long-range retained kinetic-energy levels (and sub-half-MOA accuracy) for a range of hunting calibers:
264 Win Mag 26" 156 grains RL-33 (3.8 kJ/cm^3) 3,155 fps 62 kpsi 24% efficiency
280 AI 27" 180 grains RL-26 (3.9 kJ/cm^3) 3,035 fps 57 kpsi 29% efficiency
300 RUM 28" 230 grains LRT (3.8 kJ/cm^3) 3,086 fps 60 kpsi 24% efficiency
None of these has a short, fat case. All have enough efficiency to carry over 1,500 ft-lbs of energy to an elk at distances way beyond those most of us can ethically hunt, and to get there with a fairly flat trajectory and low wind drift, maximizing the probability of hitting the target at 600 yards and beyond. And by keeping the peak pressures down a bit I can hope to extend the lives of the barrels in my 264 and 300, which have reputations as barrel burners.
The more interesting question about efficiency, in my opinion, is not the comparison across calibers, bullet weights, and barrel lengths, but the comparison for a single combination of bullet (diameter, weight, model) and barrel length--especially for the particular bullet I want to hunt. No, the 300 RUM is nowhere near as efficient as the 280 AI in the above comparison, and maybe not as efficient as a 300 PRC or 300 Win Mag shooting a lighter bullet than the 230. But among 300s able to push a 230-grain bullet to 3,000 fps, it's hard to improve much on the RUM. Shorter, fatter 300 mags (like the Norma) improve on the efficiency for lighter bullets, but their case capacity limits how much they can push the heavy 230. Shooting a 230 at 3,000 fps means I have little reason to buy a 338: most of them wouldn't give me a meaningful increment of kinetic energy or hit probability, out where it matters. They'd just kick harder and cost more to load.
In some cases a short/fat cartridge will let you use a short action (maybe a shorter barrel), saving you a quarter or half pound of rifle weight. If you're hunting and the weight matters, diet for a week before you hit the trail, and you'll lighten your load by a pound easy. Otherwise, unless you're really trying to minimize kick (especially for competitive shooting or varminting), bear in mind that efficiency is just one consideration among many. If you already have a 7mm Rem. Mag., don't buy a 7 PRC. Just buy or load the heaviest bullet your rifle's twist rate will handle at the altitude and temperatures you hunt. You might be surprised at how quickly all considerations of efficiency dwindle.