Rosh Hashanah, Chapter 3, Mishnah 4. The previous Mishnah described how they would sound the shofar and the trumpets in the Beis HaMikdash on Rosh Hashanah, while our Mishnah discusses the sounding of the trumpets and the shofar on public fast days. The fast days themselves are described in Maseches Ta'anis: at a certain point, when there is not enough rain, the community holds special prayers and fasts, and as part of this process they also sound the trumpets. In the Beis HaMikdash they would sound the trumpets together with the shofar, and this is what our Mishnah comes to describe.
The wording of the Mishnah and its details:
"B'ta'aniyos - b'shel zecharim kefufim" - the shofaros used on fast days were rams' horns, which are bent, curved and coiled.
"U'fihem metzupeh kesef" - the mouthpiece, which is not exactly where the mouth is placed but next to it, is plated with silver.
"U'shtei chatzotzros ba'emtza" - the two trumpets stand in the center with the shofaros on either side. The basis for this is the verse "with trumpets and the sound of a shofar," which uses the plural for trumpets and the singular for shofar; and even though here there is not a single shofar, the shofaros are positioned on both sides of the trumpets. The reason the trumpets are placed in the middle: they are the essence of the day - on fast days the primary sounding is with the trumpets, and only in the Beis HaMikdash did they add shofaros on either side.
"Shofar mekatzer va'chatzotzros ma'arichos" - regarding the nature of the blasts: the shofar sounds a shorter blast, while the trumpets extend their blast, so that the sound of the trumpets should be heard on its own.
"Shemitzvas hayom ba'chatzotzros" - the primary mitzvah of the day is with the trumpets and not with the shofar.
The source of this law in the Torah:
This law is based on the verse in Bamidbar: "V'chi savo'u milchamah b'artzechem al hatzar hatzorer eschem - vahare'osem ba'chatzotzros" - when you go out to war in your land against an enemy who oppresses you, you must sound the trumpets. From here we learn that in a time of distress the sounding is with trumpets, and in the Beis HaMikdash they would add the shofar to this as well.
In summary: in this Mishnah we learned the order of the blasts on fast days: shofaros made of bent rams' horns whose mouthpiece is plated with silver, two trumpets in the center with shofaros on either side, the shofar sounding a short blast and the trumpets extending theirs - all because the mitzvah of the day is with the trumpets, as learned from the verse "vahare'osem ba'chatzotzros," you shall sound the trumpets.