The Talmud states that the trumpet was made of silver while the processed horn of one of the five animal species — antelope, gazelle, sheep, goat and mountain goat — was used to carry out the ritual commandment of the sounding of the shofar (Rosh Hashanah 27a).
It also notes that preferrably the shofar should be made of a rams horn or wild goat horn, because they are curved. Rabbi Judah says, “The shofar for Rosh Hashanah must be made of the horn of a ram, to indicate submission.”
Traditionally a ram’s horn is sounded on those days because of its association with the binding and near sacrifice of Isaac (the Akeidah), which serves as the Torah reading for the second day of the festival.
Conversely, a cow’s horn may not be used because of the incident of the golden Calf (Rosh Hashana 3:2). The shofar may not be painted, though it can be gilded or carved with artistic designs, as long as the mouthpiece remains natural.
A shofar with a hole in its side wall or a chip in its mouthpiece is deemed halachically unfit, though it may be used if no other one is available (Shulchan Aruch, Orech Chaim 586).