A presto is see you soon. Arrivederci is until we meet again.

/A presto/ is the short (and usual) way to express: a) ci vediamo presto. or b) ci sentiamo presto. /arrivederci/ means: we'll see each other again. We use /a presto/ to emphasize our desire (or sometimes necessity) to see that person again and possibly soon. It could be with the meaning of a promise or of a wish, or both. /arrivederci/ is a way of saying 'good-bye' without or without emphasizing the meanings of 'a presto'. Note that friends (or people that talk to each other in 'tu' terms) usually (when departing from each other) say 'ciao', or 'a presto', while people in 'lei' terms tend to say 'a presto', or 'arrivederci' depending upon the situation and the intended meaning