There's actually two meanings.
Externally tapered are as stated. The part of the barrel that takes the most stress is towards the rear, esp. where the threads are and just in front. As the bullet goes down the barrel, pressure drops. You simply need less metal out there.
Some shooters like a "muzzle light" feel to their guns, others like "muzzle heavy". Heavy is stable, but light is faster moving from target to target. If you have enough arm strength to stabilize a given caliber/load, you're more likely to prefer a light fast-feeling muzzle.
Plus there's cosmetics
...though it's more of an "old school look". Newer visual styles involve a more "squared, machine-look".
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There have also been examples of INTERNALLY tapered barrels where the bullet gets slowly constricted. I don't know much about this but I do recall it was one of the secrets of the Python barrel, and why Python barrels were sometimes seen grafted onto S&W's ("Smolts") or Rugers ("Crugers"). I imagine it's some sort of cousin to Taylor Throating, except more extreme
. For a description of the Taylor Throat, see also:
http://www.alphaprecisioninc.com/revolver/default.htm
Basically...well, it happens I'm looking at my 38snub right this second. With the cylinder open, looking in the back of the barrel with a flashlight, there's a clear "funnel effect" ground into the back of the barrel innards. It's a total of about half a .38caliber deep. Taylor Throating lengthens and gentles that slope to about triple stock...running pretty much all the way through the barrel thread area. Normally when the barrel is screwed in, the metal tightens up right there...among other improvements, Taylor Throating eliminates this.
The Python barrel is almost one long continuous Taylor Throat, sorta
. With rifling the whole way of course, plus a conventional throat at the rear. Instead of the bullet getting slammed down to the right size in the first quarter inch, it's gently compressed over most (or all?) of the barrel length.
OR so I've read, I'm not an expert on old Colts.
