The term "non-chord tone" is used to refer to any musical note used in a melody that is absent from the accompanying chord. For example, the chord C major consists of the tones c, e, and g. Any melodic note heard against that C major chord other than c, e, or g is considered a non-chord tone because it is not part of the chord.
In essence, a tone that is not part of the accompanying chord is dissonant. Dissonances provide spice to music and thus can be an extremely effective way of creating particular emotional content in music.
And in "Yesterday", non-chord tones are used extensively to help capture the emotional and nostalgic feel for which the song is so famous. The example below (click on it to enlarge) shows several non-chord tones of the verse shaded green.
But of course, Paul McCartney is hardly the only composer to use this trick. Taylor Swift's cover of "Last Christmas", which was broadcast seemingly incessantly over the radio this past holiday season, uses the exact same pattern (same starting note against the same staring chord, with a resolution descending by step, and with lyrics displaying a similar character and degree of emotional content):