I'm taking an introductory course in Music Theory and got stuck on this part. The way my textbook explains it to me is quite confusing, so if anybody could help with a simple explanation it would be very much appreciated!
An echapee is a non-chord tone. It's defined as a tone which follows the principle tone either a step above or below then "leaves" by a skip in the opposite direction. So in a C major chord, a possible echapee would occur in the progression C-D-A or C-B-E or C-Bb-D.
I think that cambiata has more than one meaning. The one I'm familiar with is a pattern connecting a note with an ensuing note a third below; it starts with a descending second, then a descending third, then an ascending second. Example: one can connect C to A (below) using C-B-G-A. The B and G are usually on unaccented beats.
I think (but I'm not sure) that cambiata is also used for a double neighbor: C-B-D-C or C-D-B-C. (It's like a turn without playing the middle note in the middle.)
Echapee together with the Accacciatura is the two escape tones. The only real difference between the two is that the one is done on the strong part of the beat while the other one is done on the weak part of the beat.
Meaning that they are non-chord notes or notes that are foreign to the harmony that escapes the harmony and then move back towards the next harmony, it is one of the ways you can give your music color, hence the often-used term chromaticism (Chroma meaning color).
Here is an example that is taken from Wikipedia. (Take note this escape tone is on the weak part of the beat and is thus an echapee.
An echappe (note - not echapee, but with acute accents on both es) is an 'escape tone' - is a leap of a third in the direction opposite to that of the stepwise movement, as opposed to the movement back to the original note, which is called the 'returning tone'.
Nota cambiata (changing note) is a leap of a third away from an essential note. Originally a three note figure, it was expanded to four. In US it's known as 'cambiata'.
So each is an embellishment of the melody, similar to 'grace notes'.