The word 'mute' is slightly misleading because it suggests silencing. Although most mutes do indeed reduce the decibel level, their true function is to alter the instrument's tone. They capture different moods and create different textures; and they produce 'jazzy' effects.
For anyone interested in adding mutes to their kit, I will offer you my thoughts about my collection. However, please bear in mind that what suits one player may not suit another.
Trombonists and tuba-players: please note that there is a similar range of these mutes available in larger sizes for you.
I will work from the right to the left in the picture above.
(1) Rubber PLUNGER mute. Manufacturer unknown. This is one of my favourites and I use it a great deal for traditional jazz. As it is so flexible, you can squeeze and hold it in various ways and positions to achieve a huge range of effects from the mellow to the raucous and growling.
(2) Humes and Berg 102 stonelined trumpet cup mute. This is another great favourite. It gives a crispness to the tone. And you can slightly vary the effects by holding it only partially inside the bell, rather than simply jamming it in by the corks. A great point in favour of this mute is that, unlike some, it does not distort the tuning of the notes you play: it is good over pretty well the full range. No wonder this mute is so popular with many traditional jazz players.
(3) Humes and Berg 112 stonelined PIXIE aluminium mute. This is also very good. The intonation of the instrument is unaffected by it; and the tone produced is relatively mellow, so it's specially good for simply playing quietly or for 'background effects' on ballads. It also makes interesting sounds when used in combination with a rubber plunger mute, though I haven't experimented much with it.
(4) Humes and Berg 101 stonelined trumpet straight mute. This is a very efficient mute: it does just what you would expect - it allows a bright but 'different' sound out past the corks down the sides. It also keeps all the notes in tune, without distortion. It's not exciting but it is safe and is typical of what most people would expect to hear when imagining a trumpet played with a mute.
(5) Humes and Berg 120 stonelined trumpet DERBY (red and white bowler hat) mute. So far I have been a little disappointed with this, particularly because it was expensive when bought four months ago and I can't get anywhere near the sound Ken Colyer used to achieve with his famous Derby mute. I think this is partly because is has a sort of felt lining rather than metal. But mainly it's because this mute needs a lot of practice to get the feel of the many effects you can achieve, depending on the precise position and angle at which you have it, in relation to your instrument's bell. You can deflect the sound in so many different ways. I have not yet felt confident enough to use it in public. But I can tell you that it too does not distort the tuning of notes and that it certainly can produce a huge range of jazz effects. You may find - as I am doing - that it is not easy to master.
(6) The distinctive HARMON mute (complete with pull-out wah-wah stem); manufacturer unknown (but marked with a 'K'). This can be used in a variety of ways. Remove the stem and you have the silky tone associated with Miles Davis. What happens is that it stifles the familiar trumpet sound that bounces mainly off the inside edges of the bell and allows out only the sound that is left coming out of the middle. But if you put the stem in, this gives the sound a metallic tube through which to escape. You achieve a sweet, 'distant', lovely tone that is ideal for gently playing spirituals. And, of course, if you're so inclined, you can use your fingers over the end of the stem to produce semi-comical wah-wah effects. I use this mute sparingly but it certainly offers something different.
(7) BUCKET mute of a kind (manufacturer unknown). This is heavily lined with what appears to be polystyrene padding. It has a stifling, blanket-like effect on the sound. At best, you could say it produces a 'velvet' tone. I don't find this much use, so it hardly ever gets used.
(8) Humes and Berg 'color-tone' straight mute. I think Humes and Berg may have discontinued the manufacture of this mute, which is just as well, because if you want a straight mute you are far better off buying the one I described under No. (4) above. I think this 'color-tone' was made from cheaper materials. There's nothing wrong with it but it doesn't offer anything special for public performance. It would be fine as a first mute for a beginner. I use mine simply for keeping the sound down when practising within earshot of other people.
Bear in mind too that, even with my eight mutes, I am well short of the full range available.
So much for my impressions. But if you would like to have a terrific YouTube tutorial on mute usage - yes, this is really good - from a great expert, click on this video: