Old question, but I want to add to the answer...
Assuming the music is tonal or in a key, I suggest analyzing the notes of a melody in four ways:
- rhythm - metric placement
- harmony - note relative to the chord (if a the melody is accompanied)
- tonality - relative to the key (or tonal center if perhaps the music isn't clear cut major/minor key)
- interval - the relative change between notes
How to use these four aspects
Rhythm. You might use descriptions like: note is on a strong beat, note on a up-beat, melody starts after the first beat of the measure, etc.
Harmony. The note is a chord tone, note is not a chord tone, note is the third of the chord (ex. F# in a D major chord), note is seventh of the chord, etc.
Tonality. The note is the "tonic" (first note of the scale), note is flat third while the main key is major, other terms like "leading tone" could be used, etc.
Interval. you can say the melody jumps an octave, or the melody alternates back and forth on a minor third, etc.
With this as a starting point I can talk about melody using musical terms. For example: I love the Velvet Underground song I'm Waiting for The Man, because while the band is played a D major chord Lou Reed sang a flat third instead of the normal major third (F natural instead of F#.) You can hear it on the line "I'm waiting for my man" around 0:40 in the recording. In this case I'm describing the note relative to the chord. I can also make observations like most of the melody phrase start after the first beat of the measure. Now I'm speaking in terms of metrical placement. Most importantly all of these points I can make are musical choices with expressive effects.