The Torah does not explicitly mention the shofar in conjunction with Rosh Hashana. The verse only tells us
יום תרועה יהיה לכם
The key word here is teruah. However, a verse elsewhere tells us that the proclamation of the Jubilee Year is accomplished with a “shofar teruah.” Therefore the Talmudic Sages extrapolated that both of the teruah soundings in the seventh monthof the year (i.e. Tishrei) are done with a shofar (Rosh Hashana 33b).
Ram’s Horn Shofar
The Rambam writes that a shofar must be a bent ram’s horn. The Raavad and other Rishonim take issue with him, arguing that although a ram’s horn shofar may be ideal, other types of shofars, including a kudu horn, may also be used, with the exception of a cow horn.
בחצוצרות וקול שופר הריעו לפני המלך ה
Apparently this applied elsewhere in Jerusalem, but not in the rest of the Land of Israel.
A shofar that has been used for idolatrous purposes should not be used, but if it was blown on Rosh Hashana, those who heard it did fulfill the mitzvah. Likewise, a stolen shofar should not be used, but if it was, then the mitzvah is fulfilled. The reason is that the mitzvah is to hear the sound of the shofar and the concept of theft does not apply to a sound. Indeed, the blessing recited is שמוע קול שופר .
The Raavad adds that even if theft did apply to sound, the mitzvah, as noted above, is to have a יום תרועה.