The objects you’re referring to are called “mutes”. Both trumpets and trombones (as well as french horns and, rarely, tubas) can use them, and (as you’ve noticed) they change the timbre of the sound.
There are four types that I can think of offhand:[list=1]
[li]Straight mute. Among classical trumpet & trombone players, the most frequently used. Gives the horn a biting, metallic quality (especially when played loudly), but also reduces the volume.[/li][li]Cup mute. Fairly similar to a straight mute, but used less frequently. I think this generally gives a warmer sound than a straight mute, but I could be wrong.[/li][li]“Wah-wah” mute, or plunger. It looks like the rubber end of a toilet plunger because it almost always is the rubber end of a toilet plunger (at least, that’s where I got mine. I play trombone.) This mute has to be held against the bell of the horn by the player; by varying the amount that the mute blocks the bell, a “wah-wah” effect can be created (hence the name.)[/li][li]Harmon mute. Sound-wise, sort of a mix between a straight mute and a plunger. If simply inserted into the horn, it sounds a lot like a straight mute; however, there’s a hole in the centre that can be covered & uncovered by the player’s hand, creating much the same effect as the plunger.[/li][/list=1]
The first two mutes are the most common in the classical repertoire; the second two are the most common in jazz (although my jazz experience is limited; maybe someone can correct me here.)
As to the notation used: Usually, something is written above the staff to indicate where the player should insert & remove the mute (at least in classical pieces.) If you don’t mind sticking with English, just write “mute” (or “cup mute” or “plunger” or whatever) above the staff when you want it muted and “open” when you want the mutes removed. The plunger and the Harmon mutes can be a little more complicated, since the amount of stoppage can be varied (for the “wah” effect.) If it’s not indicated, the players will ad-lib it (and usually know what will sound best). I have, however, seen indications like
written over the music; this indicates that you’re supposed to be nearly closed at the note under the “+”, and fully open by the time you get to the “o”.