AP stands for Armor Piercing, which, pierces armor. It usally doesn't damage the interal parts as much as it just sails through, but if it hits something, The resulting damage is only in areas that the shell directly hit, but it can damage thicker armor targets.
HE, which stands for High Explosive does not penetrate as deep into armor as AP. It does, however, explode. If these shells hit really anything in its way, it explodes. It deals a lot more damage to open-top or thin armor tanks, however against closed-turret tanks or tanks with thick armor.
APC, which stands for armor-piercing cap, helps it penetrate deeper into armor by letting the cap take all the force from the impact instead of the tip of the shell, which increases overall penetration, and allows for thicker targets to be destroyed.
APBC, or armor-piercing ballistic cap, does the same as APC but, with a more pointed design, increased its velocity, which increase both its range and its penetration.
APHE is Armor Piercing High Explosive. which combines the best of both worlds. It both damages internal components with an explosion, and also penetrates through thicker armor with ease. The result is one dead tank and one happy gunner.
APCR is just AP but with more velocity. It does more penetration in shorter distances, but it loses more penetration over longer distances than AP. This also does has the problem of it only damaging what it directly hits, which is not the best.
HEAT stands for High Explosive Anti Tank. This does HEAVY damage to tank targets, with the chance of completely destroying the armor of a tank, which would make the tank unable to be repaired. This makes the tank completely destroye, so they could never get that tank back into battle.
HESH stands for High Explosive Squash Head, with the normal damage of the high explosive, the the squash head allows it to correct itself against sloped armor, thus getting rid of any defensive benefits of sloped armor. This makes it much more likely for a shell to kill a tank with extremely sloped armor.
APDS, which stands for armor-piercing discarding sabot, is an armor-piercing shell that discards an outer shell, revealing the smaller but much faster inner shell, which can deal insane damage. The resulting shell is just a dart moving faster than the speed of sound. This can fly through almost any tank.
APCBC, or armor-piercing capped, ballistic capped, is just an APBC shell but with an extra cap to try to improve anti-tank and naval gun penetration effectiveness.
APHEC, which stands for armor-piercing high-explosive capped, is just an APHE shell, but with a cap. This just increased the penetration of the APHE shell.
APFSDS, or armor-piercing fin-stabilized discarding sabot, is the true death dart. These fire out a shell that discards its outer shell, revealing the steel dart inside. There is NO explosive, so it only damages what it hits. But if it does hit something, it hits hard. These guys can penetrate almost ANYTHING that stands in its path, which would just leave a hole behind.
APHEBC, or armor-piercing high-explosive ballistic capped, is a APHE shell but with a ballistic cap, which again, increases its speed and penetration.
HEATFS, or high-explosive anti-tank fin stabilized, is just a HEAT shell but with some fins that keep it flying straight.
APHECBC, or armor-piercing high-explosive capped ballistic capped, is just an APHE shell, but with a cap and ballistic cap. This just increases speed and penetration again.