What is the name of the ornament played as a quick, crushed note before the main note of a melody?

The ornament referred to in the question is known as a grace note. A grace note is a very brief note that is typically played just before the main note of a melody, adding a decorative flourish to the music. It is usually not held for a full duration and is played quickly, giving the impression of a "crushed" note that enhances the expressiveness of the melody.

In musical notation, grace notes are often indicated with a small note head and are usually written at an angle to signify that they are to be played quickly before the main note. This ornamentation technique is commonly used in various musical styles to enrich the melodic line and provide an emphasis to the subsequent principal note.

The other options refer to different types of musical ornaments or techniques: a trill involves rapidly alternating between two adjacent notes, typically one whole step apart; an appoggiatura is a non-chord tone that is approached by leap and resolves to a chord tone; and a neighbor note is a stepwise motion that moves away from a main note and returns. Each has its distinct usage and effect in music, but they do not specifically describe the quick, crushed note preceding a main note as a grace note does.