The Official FFR Picture Dictionary

[LJRoX]

Although grace notes can be used as a trill see "Impromptu op. 142 no.8 by Franz Schubert." I'm playing that piece on the piano and it doesn't say tr directly below the grace note on the music itself, but below the beginning of the measure.
Arpeggios are basically chords that are broken down. Like C minor has a C Eflat and a G
you can play G C E flat C Eflat G in a broken form and call it an arpeggio.

However most trills I play are really fast and usually the trills I play are 5 notes long (eighth with 4 sixteenths slurred together). Also trills can be composed of 3 different notes, not always the regular two
like C D C D C D C B C, depending on what key you play on first.


Edit: the thing is called a turn or an ornament.. I wasn't too sure about the name.