Question

What are all the moves and abilities that have different effects based on the targets/users/ally's type?

Comments

Curse and (arguably) Revelation Dance are two that immediately come to mind, though there's probably more I'm just forgetting.

If we expand to include items, Black Sludge can be added, too.

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Almost all damaging moves

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effects, not damage.

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damage is an effect, no?

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No, effects are things that have to be specified under the damage it is said to deal, eg Mud Slap does damage, and has an effect, to lower accuracy.
Additionally, even if I'm wrong, I don't think OP was asking for every typed damage dealing move, just ones like Rototiller, Curse, Symbiosis, etc where it has to be specified that it changes.

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nobody?

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This comment ignores the type chart and specific immunities, although it's probably worth mentioning that glare and thunder wave are the only status moves that have ever used the type chart (and thunder wave is the only one that still uses it).
all attack moves that cause major status effects (gen 1 only)
conversion (gen 1 only)
toxic
transform
curse
foresight
hail
odor sleuth
present (gen 2 only)
sandstorm
ingrain
sheer cold
gravity
miracle eye
roost
fire pledge
grass pledge
reflect type
smack down
flower shield
freeze-dry
rototiller
burn up
revelation dance
thousand arrows
double shock
nihil light
snowscape
tera blast
scrappy
imposter
corrosion
mind's eye