The word "arpeggio" is Italian, meaning "in the style of a harp."
Imagine someone playing the notes on a harp one after the other and you'll understand exactly how an arpeggio should sound. Like many forms of piano accompaniment, the arpeggio is based around the chords or triads popular in most styles of Western music.
Here's a picture of the keyboard and the C chord again. Play the chord with each hand just to refresh your memory. The red dot represents middle C.
