The Gemara informs us that it is a mitzvah on Rosh Hashana and Yom Kippur to use a curved shofar because it reminds us to bend ourselves (i.e. bend and submit ourselves, rather than being stiff-necked and un-repentent). The remainder of the year a straight shofar is best (Rosh Hashana 26b).
Another reason to prefer a curved shofar is that it serves as a remembrance of the ram whose horns got tangled in the brush following the Binding of Isaac (Akeidas Yitzchak) and which was later sacrificed in place of Isaac.
The Ran writes that it is a mitzvah to use a more curved shofar, but it is not mandatory. The Rambam (Hilchos Shofar 1,1), on the other hand, holds that the shofar must be curved in order to fulfill the mitzvah.
ושופר שתוקעין בו בין בראש השנה בין ביובל הוא קרן הכבשים הכפוף
The Shulchan Aruch seems to side with the Ran.
If one is faced with a choice of using a curved shofar with mediocre sound or a straight shofar with very good sound, HaRav Nissim Karelitz writes that the curved shofar would be preferable. The reason, he explains, is that according to one opinion curved is required, whereas according to all opinions nicer sound is no more than an embellishment of the mitzvah (הידור מצוה).