If we're talking about ships, a common way to understand the advantage in an engagement is simple arithmetic to figure the "broadside weight" or "weight of metal;" in other words, counting up the combined weight of one side of the ship firing all of its guns.

Let's take for instance a 28 gun frigate. Twenty-four of its guns are broadside guns, aligned on its gun deck to fire perpendicular to the motion of the ship. Four are "chasers," mounted in the bow and stern. The number and type of each gun should be suited to the ship's ability to not only bear their weight when sailing (each gun might weigh several hundred pounds at least), but also how well the ship itself could withstand the firing of its guns: the recoil of a large gun could damage the gun's mounting and could even cause damage to the planking and structural supports of the ship itself.

So, arming the ship, we have to consider the ability of the ship to bear the weight and recoil shock of the guns it carries, and a somewhat decent broad framework for this is simple size. In our example, a 28-gun frigate is a single-deck, relatively small ship, and would be unlikely to be able to bear twenty-four 32-pound long guns. Let's take, for example, the Royal Navy's "Enterprise" class sixth-rate frigates. They mounted twenty-four 9 pounder guns on the gundeck, and carried four 6 pounders on the quarterdeck as chasers. One of these frigates could fire, if all guns were mounted, a broadside weight of 216 pounds.

If the ship could bear it, we could scale up the damage-dealing broadside guns to 12 pounders, bringing our broadside weight up to 288 pounds. If we were really ambitious and the ship was well maintained, we might even get away with 18 pounders, for a broadside total of 432 pounds.

Keep in mind this is just a loose metric that could compare the ships in their ability to deal damage, instead of the myriad other ways that ships were categorized and sized up: the confusion of frigates and sloops and brigs and snows and razees and ratings, to say nothing of sail plans and nuances of design and construction are all factors in a naval engagement, and good captains were generally relied upon to know in an instant the tonnage, size, speed and maneuver, armament, and broadside weight of their opponent's ships with some degree of accuracy - these factors are often used in courts-martial for captains who lost their ships, as well as in the more day-to-day despatches and descriptions of ships taken. For example, Horatio Nelson reported the engagement of the HMS Minerve, under the command of Captain George Cockburn, against a Spanish frigate:

Captain Cockburne brought his Ship to close Action at Twenty Minutes before Eleven, which continued without Intermission till Half past One, when La Sabina, of Forty Guns, Twenty-eight Eighteen-Pounders on her Main-Deck, Two Hundred and Eighty-six Men, Captain Don Jacoba Stuart, having lost her Mizen-Mast, (as he did after the Action) Main and Fore Mast, One Hundred and Sixty four Men killed and wounded, struck her Colours.

Attention was paid to the class of the ship, its armament, and its crew count. The Minerve herself was a 40-gun frigate, which carried twenty-eight 18-pounders and twelve 8-pounders, a broadside weight of 408 pounds. The Spanish frigate, was similar in size and armament, and only going on Nelson's report of the twenty-eight 18 pounders on her main deck, would have given each side of shot at 252 pounds.

In terms of the raw power of the shot sizes, it seems fairly clear that in frigate actions, 18 pounders were considered heavy enough to deal damage but light enough to be carried well on smaller ships. Ships of the line, the real big ships, might carry hundreds of guns of a range of weights, but it wouldn't be out of the realm of possibility to carry 36 pounders or more.

The arithmetic of this is all based on being able to effectively damage the enemy ships: it seems like 9 pounders were, for frigates, about the lightest we might want to carry as anything but chasers, and as the wars of the French Revolution wore on, the trend was for heavier and heavier main-deck guns, all the way up until all the math is thrown out when carronades make their entrance.

Carronades were nothing much more than truncated cannons; shorter, stubbier, but heavier built so they were less likely to explode, but they were also allowed smaller ships to carry much heavier guns; the Minerve eventually supplemented her 18 pounder main-deck guns with sixteen 32 pound carronades, and switched out the 8 pound chasers for 6 pounders. Some quick math puts the Minerves carronade broadside, just with the 18s and 32s, at 508 pounds.

To round this all out: the heavier the ball, the more damage it could possibly inflict, but the more weight of the gun to fire it, the more weight of powder to propel it, the more severe the recoil, and the more chance to throw the ballast of the ship out of whack. Ship's timbers functioned as a sort of armor, but also protected them from their own broadsides.

But the weight of shot, while not the singular element of single-ship action success as championed by Teddy Roosevelt (he wrote The Naval War of 1812 that stressed broadside weight as more or less the reason the Americans won so many single-ship actions, and says that it was the determining factor in the Battle of Lake Erie), was absolutely an important factor in ship engagements, and we can see a steady creep in the weight of armament of even similarly-classed vessels throughout the duration of the long 18th century.

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Sources as linked above.

Sorry for the late response. I just have a couple more things to add to u/partymoses' answer.

In addition to the weight of shot there are a couple of other ways to potentially compare cannon effectiveness with various degrees of applicability. One of the other big ones is the weapon's range, particularly its point blank range. Because of the cube-square law a larger cannonball of a given density will have a greater mass relative to its surface area/cross section and will be slowed less rapidly by air resistance than a smaller ball at any given velocity, usually making it possible to achieve a much flatter and more consistent trajectory, in addition to retaining much more of its kinetic energy over time. Both at sea and on land, accurately judging a targets range on the fly remained extremely difficult if not considered almost impossible much of the time, so flatter trajectories and having a greater distance at which you could just point the cannon straight at the target without having to worry too much about the exact range tended to correlate with much better accuracy, especially in combat. This tended to be the one downside with large caliber, low-velocity guns like carronades. Although in theory they might be very energy efficient if fired at a high angle and even fairly consistant if the windage wasn't too wide (i.e. relatively good precision if not accuracy), the short point blank generally limited them to being used as short range weapons in ship to ship combat. Often ships with carronades would carry for them both solid shot and hollow round shells, which, though ballistically inferior to solid shot in just about every way, could be fired out of a carronade with a higher velocity and greater point blank, making them more useful at longer ranges. In Naval warfare by the late 1700s there was generally a growing emphasis on "battering" over overall power or penetration, i.e. even if it hit a wooden at a lower velocity, a larger cannonball would still tend to cause a much larger, more jagged hole which was more difficult to repair, which helped make overall weight of shot still the more useful metric for ships.

There are a couple of other factors which could influence a gun's overall effectiveness in addition to shot weight and velocity, including rate of fire and ability to penetrate fortifications. A larger bore also meant that a gun could fire a larger quantity of grapeshot/canister or could fire a larger explosive shell as the quality of fuses improved. You could also from time just find experiments with guns being fired repeatedly on a range at paper targets being used as a metric. A source that might be of interest to you is William Muller's 1811 Elements of the Science of War where starting page 136 of the second volume he includes a long section discussing the effects of artillery based on a whole bunch of data he'd collected during his years teaching at a Prussian military college. Page 151 he provides data on how far various sizes of gun will penetrate into earth or masonry at different ranges. Then on page 165 he puts a whole bunch of data together to give a theoretical estimate of the overall casualties inflicted by either a 3-pounder, 6-pounder, 12-pounder cannon, or 5.5 in. howitzer against an advancing enemy line. Concluding that the 6-pounder and howitzer were about equal, inflicting about 1.5-1.7 time as many casualties as the 3-pounder, and that a 12-pounder should inflict about twice as many casualties as a 3-pounder overall.

The if you want more, the first volume includes a bit more information on the technical side of things such as the specifications of various guns, carriages, recommended crew complements, and the recommended number of horses for transport.