I seem to get easily confused between the 3 terms. Is an appoggiatura a grace note? How can I distinguish each of them in a music score?
The appoggiatura ornament indicates a resolution of a suspension and does not have a stroke through it. They induce a feeling of "yearning". The notes take actual time in the measure relative to what note type is used to represent them. If there are multiple notes in the ornament, they should all take time from the principal.
The acciacatura is the traditional grace note with a slash through it, typically indicated by a quaver. These notes are supposed to take no time and serve to ornament whatever pitch they are on. This is true for any amount of notes in the ornament.
The images are taken from Wikipedia so they should fit under the Wikimedia license.
I can't add anything to the other answers that say what the notation actually means.
But what I can add is a suggestion for how to remember which is which:
appoggiatura sounds like the word podgy so its notes are fatter. So the other ornament, acciaccatura must be the ornament with the crushed rhythm.