The Alto Trumpet is rarely seen anymore. It is pitched in either E-flat or F a sixth or a fifth below the standard C Trumpet. In many ways, it is the exact instrument that was used in the Nineteenth Century for their F Trumpet parts, but with a slightly bigger bore. This bigger bore favors the lower notes.
There is one major piece in the orchestral literature that calls for the Alto Trumpet: The Rite of Spring. Here, I am going to clear up a huge problem that every single orchestration book has gotten wrong. Stravinsky wrote for his E-flat Alto Trumpet (called Bass Trumpet in the score) down to a written low F natural. The instrument cannot do this. Period. Orchestration books have written that instruments will have another valve in order to reach this note, but they fail to see what the real problem is. No one wants to admit it, but Stravinsky made a mistake. The instrument cannot play the note, so the part, a highly exposed trumpet ensemble section, is never played on an Alto Trumpet, but rather on the bigger Bass Trumpet that can play the written low F (sounding a low A-flat). Some modern 4-valve Alto Trumpets now can play this note (and notes down to a written low D), but these cannot be counted on as they are rare.
Stravinsky’s teacher, Rimsky-Korsakov tried to add the Alto Trumpet to his opera scores, but that great master orchestrator defeated his purpose by never utilizing the instrument as it should be used. Rimsky-Korsakov kept his Alto Trumpet parts in exactly the same register as his B-flat or C Trumpets, and never took them lower than the standard range.
Today, if we do see an Alto Trumpet it is in a larger trumpet ensemble. Here, its added range is needed and is fully exploited. Unlike the Bass Trumpet, the Alto Trumpet is almost always played by a normal soprano brass player (i.e. a trumpet player).
With most Alto Trumpets, the sound is not much different than a standard B-flat Trumpet. If anything, it is slightly darker in sound. The main advantage to the instrument is the added lower range (an extra fifth lower). It could make for an exciting and dramatic fanfare in a slightly lower tessitura.
An example of an E-flat Alto Trumpet. Note the larger size of the mouthpiece compared to a normal trumpet.