I sometimes see or hear pairs of these words confused.

Problem:
These three nouns are not synonyms.

Explanation:
The noun “allusion” means a casual or incidental mention of or reference to something. The key part of this noun comes from the Latin verb “alludere” (to play with), from “al-” (toward) and “ludere” (to play).

The noun “delusion” means a false opinion or belief. The key part of this noun comes from the Latin verb “deludere” (to mock or deceive), from “de-” (down) and “ludere” (to play).

The noun “illusion” means something that deceives. The key part of this noun comes from the Latin verb “illudere” (to ridicule or mock), from “il-” (in) and “ludere” (to play).

I sometimes see or hear “allusion” and “illusion” used interchangeably (even though they should not be), apparently because the two words sound similar to one another.

I sometimes see or hear “illusion” and “delusion” used interchangeably (even though they should not be), apparently because writers or speakers see “ill”, which looks negative and somewhat like “de”, instead of “il”, which is the actual prefix of “illusion” and is simply a variation of “in”.

Note that one can have a delusion because of an illusion.

Solution:
Use “allusion” when one means reference. Use “delusion” when one means mistaken belief. Use “illusion” when one means deception or magic trick.