There are three types of mutes for the horn - a stopping mute, a straight mute, and a practice mute.
Stopping mutes are often made of metal and are somewhat pear-shaped. Its use will have the same effect as completely closing off the bell with your right hand ("hand-stopping"), which is generally preferred over using the mute. It is very important to note that this will cause the note to jump up a half-step, so you will need to finger down a half-step. If, when hand-stopping, you find that the tone jumps down a half-step rather than up, it is because you are not completely closing off the bell. Make sure that there are no gaps between your fingers. It is not uncommon for there to inadvertently be a gap between your thumb and your index finger. If you want proof that jumping down a half-step is not correct, try playing a concert E arpeggio with open valves, hand-stopped. A + sign over the note indicates to start using the stopped horn (either hand stop or use the stopping mute) and o means to return to regular, unstopped horn. In other words, + means stop, and o means go.
This type of mute does not require as much explanation. There is no alternative to the straight mute in music that calls for it. Straight mutes are usually cone-shaped and made of wood, but some are made of metal. con sord. (short for con sordino, Italian for "with mute") means to insert the straight mute. senza sord. (senza sordino - "without mute) means to remove it. Most straight mutes have wrist straps. It is strongly recommended that you hang the mute on your right wrist before beginning a piece that calls for it.
Practice mutes are never used in concerts, thus their name. They simply make your sound much, much quieter to avoid disturbing others when you have to practice in hotel rooms, etc.
With any type of mute, it is a good idea to twist it slightly when you insert it into your bell. This will make it more secure and prevent it from falling out.