My conductor at university loved to bring up the story of one composer who specified what mute the string section should use in performances of his piece. What this tells us is that different mutes impact the violin or cello differently, and perceptibly so!
Mutes are generally made of plastic or metal and attach to the bridge. There, a mute dampens the sound of your string instrument by limiting vibrations. Some mutes, usually orchestral mutes, just bring everything down a level. Other mutes, usually classified as practice mutes, may severely muffle your sound to the point where you are left almost with plain pitch and no depth.
When choosing a mute, your primary considerations should be, in this order, sound quality, usability, and aesthetics. If the mute does not produce the sound you need it to, it is the wrong sort of mute. If the mute cannot be moved on and off the bridge easily, it may not be the mute for you. If the mute is remarkably ugly or garish or too plain, perhaps you should look for a different mute.
Please note: As someone who is primarily a violinist, most of my remarks on mutes are in the context of the model that is used on violins. On other instruments, I have only used the basic Tourte mute long-term.
Mutes and Short Reviews
Alpine Mute Co. Professional at Southwest Strings
Notes: Felt stiff at first. Easy to slide on and off. Sound was not discernably unique enough from lower-priced mutes. Not worthwhile if you have another mute that produces similar sound.
Mouse-Tro at Southwest Strings
Notes: Poorly designed and poorly made. Extra bits of plastic. Touches string on the wrong side instead of only sitting on bridge.
Practice Mute, Metal
Notes: Like the rubber practice mute, this heavily mutes your instrument for practice. I do not like having something that could scratch my violin on my violin, personally. It is all too simple for a metal mute to slide off a wooden bridge.
Practice Mute, Rubber at Southwest Strings
Notes: If you do not have complaining neighbors, over-the-bridge mutes of this sort are unnecessary. These are to dampen sound enough that you are less annoying to others.
Spector Mute at Johnson String
Notes: A little different, but it does produce a different sound. Need to get used to a slightly different sliding motion, but that is simple. Comes in black and copper, for some reason.
Tourte Two-Hole Round Mute at Johnson String
Notes: Some are very stiff, which I do not like the tactile feel of and which produce a sound I like less. The softer, pliable ones tend to produce a sound I do like. Is it that the rubber is too old, for the stiff ones?
Notes: My favorite mute is also inexpensive. Very easy to slide on and off.
Wire Slide-On Mute at Amazon
Recommend: Not for me.
Notes: Although this does produce a unique sound, it is possible to damage your strings if you are not gentle enough when using this. That risk is enough for me to be wary of it. However, many people like how this mute style sounds and how it works.