I call them grace notes when they're notated as such, which works if you're talking to people who know that early-period grace notes aren't always interpreted in the modern sense of before-the-beat. With other people, you might have to keep saying appoggiatura.
The verb "appoggiare" means to lean. An appoggiatura leans on or presses against the main note and the prevailing harmony. As with other aspects of music, its function went back and forth as style periods changed. In medieval music, it had a melodic function; renaissance, somewhat more harmonic use, although suspensions were used much more than appoggiature; in early baroque it was back in melodic use, but in late baroque more as harmonic accent or embellishment; since then, it's come back to melodic use (as in the rondo alla turca) and fully written out instead of as an ornament.
It can be represented as a grace note or be fully notated, and can come from above or below-- the dissonance is what makes it an appoggiatura, not the way in which it's notated. When shown as a grace note, the rhythmic division can be interepreted many ways, depending on the time period, genre, composer's style, and mood of piece: for a Bach fast movement, maybe the grace note would be half the length of the main written note, or even shorter/faster, whereas for a Sarabande with a grace note before a dotted half, you could make the grace note 2 beats long and the main note just 1, which is very poignant and expressive. Generally, I like Bach appoggiature longer than shorter, although judging from most editions, I may be in the minority.
There have been many attempts to standardize ornament notation, and one of the rules which evolved along the way was the slash/non-slash to differentiate before-the-beat from on-the-beat. There are various reasons for why it was notated as a grace note. Partly it followed rules of harmonic writing by writing consonant notes as "main" notes and strong dissonance as expressive embellishment, also to encourage the standard ornamentation of the time and region, and, I understand, sometimes to keep the performer from further ornamenting a particular spot by making the desired figure read as an ornamentation already (I think that happened somewhat in the late baroque, at least I've been told).
Composers have been wildly inconsistent, even with themselves, about how they showed appoggiaturas and other ornaments, so you can't completely trust the slashes or their absence. Editors use the slashes, and with a good edition, it's worth giving the markings a try and see how it sounds. But a very reliable source of information and taste is the composition itself, because usually there's something in the writing that gives clues about the ornaments. For example, many composers will fully notate a figure in the very beginning that will come up again a hundred times in the piece, but later on uses ornament symbols as shorthand to imply the same figure, and you have to look carefully to realize that he's asking for the same thing with a different spelling-- Haydn does this. Also, if the composer has written choral works with accompaniment, the voice parts tend to have appoggiaturas fully notated so the choir can stay together, so you can compare that against the instrumental parts which use more ornament symbols, and learn about the composer's thoughts that way.
Ultimately the performer needs to experience how various ornaments sound and feel when done different ways in different kinds of music, and see how great composers wrote them out, when they did so, and then make artistic decisions from there.
(sorry, this got long)
Topic: A question about appoggiatura