Answer: The appoggiatura is a "non-chordal" or "nonharmonic" tone - a note that is not part of the harmony at the moment it sounds, but rather is a second above a harmonic tone. Speaking without regard to chords, as in pure counterpoint, the appoggiatura is a dissonance - forming a dissonant interval with another sounding tone - that is approached by an upward leap and is left by a downward step.

In music of the classical era you'll generally see the appoggiatura written as a small note - a grace note - that is followed by a note a second lower and is slurred to it. But sometimes, especially if an editor has intervened, you might see the appoggiatura appear written out as it would be played, as a normal-sized note slurred to a note a second lower and having the same time value.

That sounds sort of abstract: here's an example of an appoggiatura written as a grace note and one written out as it would normally be played, with the grace note having half the value of the main note:

An appoggiatura is always "on the beat" or it doesn't deserve the name. If you see two notes that form a descending second, and the first is in an accented position and is approached by an upward leap, that is possibly an appoggiatura. If the accented note is dissonant, moving to a chord tone before the harmony changes, that confirms its nature.There is an audio example in the Wikipedia article on the appoggiatura in which you hear what is claimed to be an appoggiatura but it moves upward by step. This detracts somewhat from the personality of the creature. The true appoggiatura is a metrically accented note approached by upward leap and moving downward by step to a consonant tone of the continued harmony. If it were approached by downward leap and moved upward it could be called an "inverted appoggiatura," and if approached by step in the same direction it would be an "accented passing tone." If it is approached by leap and left in the same direction it is just plain wrong and deserves no proper name among decent people.