For the majority of vehicles, "normal" AP is the best all-rounder for killing tanks. It penetrates quite well, and explodes inside targets to deal damage. There are some variations (APHE, APCBC), but the difference is very small. Loses penetration over range.

Keep in mind that the German ammo doesn't say APHE, but it still has explosive filler. Some rounds, however, especially a lot of American ones, don’t have explosive filler, essentially functioning like a worse APCR. See below.

Now for HE (high explosive). It’s pretty straightforward, just a bomb that explodes on contact, and the value of penetration on the card is for each fragment that comes from the explosion.

For most tank vs tank combat, HE is pretty useless. It’s only intended for targets that are thinly armored or open-topped, like Flakpanzers. However, for very large guns (122mm and above) HE is great, because you can simply shoot the turret or the very top of enemies and the fragments are so powerful that they go through the top of the hull and wreck shit.

HEAT works basically like this: when it hits the target it explodes, launching a stream of metal into the enemy tank. Does not lose penetration over range, and is countered by spaced armor. It's mostly for short barreled guns like on the early Pz IVs, and the first StuG, simply because their AP ammo is crap (low velocity on the gun).

Now APCR/APDS. It's basically a solid penetrator, like a spear that goes through the tank; it does not explode inside targets, although it’s possible to create some spalling upon impact. It has better penetration than standard AP, and is quite powerful in some situations, although for extreme ranges, it loses penetration faster than the normal AP due to its smaller weight. It also travels faster, meaning you don’t have to lead your shots as much.

The German 50 and 75mm guns, with their poor explosive damage on the normal AP, can use APCR to great effect, because you can hit a crew member, and the round will keep going and kill the guy behind him. Against some tanks, you can kill 3 crew members with one shot in this manner.

Remember: more penetration doesn't always mean better ammo. Sometimes, but not always.

Remember to carry only around 40% of the max ammo (depending on tank). This leaves certain ammo racks empty in the tank, reducing the chance of an instant kill. Consult this, courtesy of /u/Strikaaa.