First you should understand that chain shot was primarily used by naval ships attacking other naval ships. In the age of sail powered vessels, it was fairly effective. The point of chain shot was to cut or tear the rigging ropes that were used to work the sails, or ideally the standing rigging that actually held the masts upright. Causing an opponent's masts to fall, also called dismasting them, gave one a large advantage in naval combat. The disadvantage was that chain shot had higher air resistance and thus shorter range than regular roundshot. It was also somewhat more expensive and slower to load.

Fast forward to the American civil war. In the civil war era most ships fought under steam power, rendering sails and thus chain shot irrelevant. For the most part civil war era warships did not carry chain shot.

There were some experiments with using chain shot in land battles. Most famously, John Gilleland's double barreled cannon. Gilleland made a cannon, which still exists, with two barrels that angled slightly away from each other. His notion was to load each barrel with a separate ball and an attached chain stretching in between. He figured the slight angle between the barrels would cause the chain to stretch out horizontally and mow down enemy soldiers like wheat before a scythe. Test firing revealed, however, that this never happened. Instead, one barrel would always fire slightly sooner than the other, one end of the chain would fly out while the other was still deep in the other barrel, and the chain would whip around in the vicinity of the gun. Used as intended, the weapon was far more dangerous to the gun crew than the enemy. Chain shot never caught on as an ammo type used in land battles, as simple canister (basically a giant shotgun shell) was more effective.