On a single action revolver, the quintessential "old west" or "cowboy" gun the trigger does only one thing: release the hammer, causing it to drop and hit the bullet primer, firing the gun.
To fire a second shot from the same gun the shooter will need to pull the hammer back with their thumb. When pulled all the way back the hammer will hold itself in the cocked position. The hammer being pulled back will rotate the cylinder so that the next chamber is under the hammer. Pulling the trigger will now fire the next round in the cylinder, until all six (only five are usually loaded on old guns, as another safety precaution to prevent the hammer from resting on a live primer). The most famous single action revolver is the Colt Single Action Army:
The other type of revolver, and the one that was widely used by law enforcement from about 1900 to 1985 (when they began all moving to semi-automatic pistols, and away from revolvers) is the double action revolver.
The term double action refers to the two actions pulling the trigger accomplishes: a long (about 3/4") trigger pull cocks the hammer, rotates the cylinder AND fires the pistol by releasing the cocked hammer, all in one smooth motion.
This was considered a big advantage. The shooter no longer has to manually cock the hammer between every shot, merely pulling the trigger over and over will cause the gun to fire all six (or however many) rounds are in the cylinder.
The classic double action revolver is the Smith and Wesson, of which there are hundreds of different models all following the same general design. Here is a 4" Model 686, a very popular police revolver chambered in .357 Magnum/.38 Special.
There is one little twist with the double action revolver like the Smith & Wesson: it can still be fired like a single action. Most (but not all) double actions the shooter could still thumb the hammer back until it locked in the cocked position. This would have the effect of moving the trigger back in the guard about 1/2 and inch, reducing the travel needed to fire the weapon from the full 3/4" in double action mode to a mere 1/4 inch in single action mode, with a commensurate decrease in trigger pressure, from about 12 lbs in double action, to about 4 lbs in single action mode.
Police were taught to always use the gun in double action mode. That (as you point out) is much more effective for defensive shooting. Only one simple motion to master: the long double-action pull. It was also considered much less likely to result in an accidental discharge (or negligent discharge, and some prefer to call them) - because of the heavier and longer trigger pull.
Handgun hunters often prefer to fire them in single action, to get the shortest, lightest trigger pull, which helps with long-distance targets.
The other bid advantage of the double action revolver is that the cylinder swings out and all six chambers can be emptied of spent brass with a single movement of the small ejection rod. It can also be loaded much faster than the older cowboy style gun.
Despite being an antiquated design the Single Action style lives on. Some of it is the historical vibe they give off. Others, mostly hunters, don't need the self-defense double action capability of the more modern Double Action style, and prefer the strength and simplicity of the old design, which lends itself particularly well to magnum loads.