If you play jazz, this is a necessity as it isn't unusual to at least need this for one song on every gig, if not many more. Usually this is done for effect such that you will see, for example, one note with a + and the next note with an 'O' creating a 'wah-wah' style sound. So if you see a plus sign, it will mean put the mute over the whole bell. You will also frequently see symbols 'o' and '+'. You won't totally close off the bell, but - as the second name indicates - cover it at around a 45 degree angle such that about half of the total direction that the sound can go is blocked.

When indicated, what you will do is hold the plunger with the part that would hold the stem between two of your fingers with the bottom part facing the bell. It is literally a plunger, like what you would use to unclog a toilet, minus the stick. Plunger or 1/2 (Half) Plunger - This mute is just like it sounds. However, most of the time, a particular type of mute is indicated. When it comes to commercial music, mutes are not generally as expensive but it isn't unusual to carry around at least five mutes if not more. Jo-Ral and Dennis Wick are common brands of this type of straight mute. In this setting a metal or wooden (not cardboard though) mute is typically desired. If a mute type is not stipulated, the vast majority of the time you should use a straight mute. In traditional, classical music and other chamber style music you may see the term Con Sord or some similar Italian or even German markings (Con Sordino, Mit Dampfer, etc.). For trombone you'll typically see a number of mute markings, but the style of music changes what you'll frequently need.

Soft tone mute trombone