Shooting without hitting is no fun and hitting is easier with a 20 than with a .410. Give the kid a chance and get him a 20, a .410 is the expert's gun. The 12 bore loaded with slugs is the best choice for bears, but a 20 ga slug is sufficient.
If you shoot at a large paper target with a load of bird shot, the paper will be perforated with dozens of holes from the pellets passing through it. The holes in the paper mark the pattern, and the diameter of the pattern at any given range can be controlled with the choice of appropriate choke. Think of chokes the way you think of an adjustable the nozzle on a garden hose. The greater the constriction of the nozzle, the farther the water will travel from the nozzle, the less the constriction the wider the spray, but the shorter the distance it will travel. So it is with shotgun chokes. Different chokes offer different degrees of constriction. When the pellets leave the muzzle of the shotgun, the pattern is very tight because it is the same size as the muzzle of the gun barrel. But the pattern becomes cone shaped as it moves down range and grows in diameter. A very wide pattern from a cylinder or open choke makes hitting easy, but because the wide pattern doesn't last very long so the range is short.
A cylinder choke is an open choke with no constriction at the barrel. This choke has often proven to produce the best accuracy for shotgun slugs, although they can be safely used through most chokes, and some prefer an open choke for large diameter buckshot which does not respond to choke constriction in the way small diameter birdshot does.
The modified choke has a moderate amount of constriction, so the range is extended beyond what is possible with the open choke. It will be more difficult to hit very close targets with the modified choke because the pellets do not spread as soon as from an open, so greater range is attained. The modified choke is a very useful choice for the gunner who primarily uses birdshot.
The full choke offers the greatest range due to its greatest degree of constriction at the muzzle. This is the choke most often chosen by waterfowlers, although some have found that modified chokes give better performance with steel shot.
Many of today's shotguns come with a adjustable or replacement chokes which gives you the opportunity to test your ammunition with each choke to determine which best suits your needs.