The Gemara in Rosh Hashana (34a) seems to offer 3 possibilities.
1. What we call a תרועה
2. What we call a שברים
3. What we call שברים-תרועה
The Gemara there quotes an interesting statement from ר' אבהו that he instituted the practice of blowing all 3 because of a ספק which is the correct תרועה.
The Rishonim ask a very fundamental question. What happened before ר' אבהו? Before ר' אבהו, some communities only blew שברים and some only תרועה. Were they not יוצא? How did this ספק come about? Everyone hears shofar every year so how did they forget what it was?
R' Hai Gaon has a fascinating answer. He says that really all 3 are called a תרועה and you are יוצא the mitzva by blowing any one of the 3. תרועה means a crying type of sound and all 3 fit the bill. ר' אבהו was bothered by the fact that since different communities blew differently people would not understand and think that the other person is not יוצא, he also felt that it made it look like there are different Torah's. Therefore, he instituted a common practice of blowing all 3 different possibilities. This is how the Ritva and others explained that in Musaf the minhag was to blow תשרת for מלכיות and תשת for זכרונות and תרת for שופרות. If there really was a safek then you should need to blow all 3 after each section of the שמונה עשרה. However, according to R' Hai it makes perfect sense, since you are יוצא with any of them we are not מטריח the people to blow so many קולות.
Tosafos and the Rambam however disagree. They hold that ר' אבהו had a real safek and you are only יוצא with one of the 3, the other 2 are wrong. It comes out, that before ר' אבהו it would seem that many people were simply not יוצא the mitzvah. The obvious question is how could this have happened? In truth, we can ask this on many cases. For example, everyone wears tefillin every day and yet there is a machlokes what is the order or the parshiyos. You have to say that over teh years the mesora was lost or got corrupted and there was a genuine doubt as to what is called a תרועה.