Mutes change a trumpet's tone and volume by altering how sound radiates from the bell, and the three most common types each do this differently.
Straight mute
Cone-shaped, held in the bell by cork strips, made from plastic, aluminum, copper or cardboard depending on the model. It is the most commonly required mute in band and orchestral parts and is usually the first mute a student needs.
Cup mute
Similar shape to a straight mute but with a wider cup on the end. It produces a quieter, more muffled tone that dampens higher overtones, giving a softer, warmer sound than a straight mute - common in big band and jazz ensemble writing.
Harmon mute
A two-piece aluminum mute with a bulb-shaped body and a removable stem, sealed against the bell by a cork or rubber ring rather than just resting inside it. With the stem in, it produces the buzzy, metallic tone associated with jazz trumpet; without the stem, or with the stem pulled partway out and the hand cupped over the end, it produces a different, more vocal effect. It is a jazz-specific mute rather than a standard band-repertoire requirement.
What this means for buying
Mutes are sold separately from the trumpet and are not something to expect included with a used horn - a beginner generally only needs a straight mute to start, with cup and Harmon added later depending on what repertoire actually calls for them.