From what I have heard there were those that did not eat turkey (such as Rabbi Yaakov Kamenetzky Zatzal) due to a question whether we have a Mesorah as to the Kashrus of this bird. I know that many people do eat turkey, myself included. What are the two sides of this dispute? Is there really no Mesorah, then how come many people eat it? If there is one then what is the reason that some people do not?
Here's a clear explanation of the Heter side:
The Arugot Habosem (Rabbi Aryeh Lebush Bolchiver, author of Shem Aryeh, Russia, published 1870; kuntras ha'tshuvot in the back, siman 16) very neatly presents the quandary: Birds require a tradition to be kosher and turkey (indik) is a bird that comes from America, a place that was not discovered until the year 5254 (1494) and so no tradition is possible. 55 Yet, he notes, all Jews, except for one well-known family in Russia, the Frankels, eat it. He therefore reaches the very important conclusion that when the Ramo requires a tradition, it is only when there is uncertainty about the bird's dores status. He posits that if a bird is no longer "new" but has been observed for a long period of time, i.e. 12 months, and observed to be non-dores it is possible to say with certainty that it is not a dores. Furthermore, he rejects the Avnei Nezer's (YD 1:76:13-14) contention that domestication and living among "friends" may be the reason for the non-dores behavior. Arugot Habosem therefore posits that even according to Rashi and the Ramo, a bird, like the turkey, that is raised in thousands of houses for hundreds of years and is clearly not a dores does not require a mesorah if it also has the three other indicia of a kosher bird; which the turkey does. 56
Taken from here.
It's pretty clearly widely accepted, but note that there are still those, today, even in what is considered the Modern Orthodox circle, who don't eat turkey. One prominent example reported to follow this (although I cannot locate a source online at the moment) is R' Hershel Schachter of Yeshiva University, who is one of the Poskim for the Orthodox Union.
I have confirmed with a family member of R' Schachter's that he does not eat turkey because of a lack of Mesorah, but I didn't want to pry and ask if that's his official Pesak when others ask him, since I could just ask him myself but don't feel compelled to. The article linked above, however, cites Nefesh HaRav, written by R' Schachter, and quotes R' YB Soloveitchik as permitting turkey.
We do have a masorah on turkey. That is:
Bechorot 7a declares that kosher and non-kosher species cannot cross-breed. Thus, if two species can hybridize, and one is known to be kosher it is proof positive that the other is kosher as well. This is cited (Rambam, Ma'achalot Assurot 1:13) as an halachikally valid means of distinguishing between kosher and non-kosher animals, and should obviate any need for a mesorah, when it can be applied.
But I've seen this presented as a masorah of sorts. Namely, we have a masorah on chickens. And we have the talmudic statement. Therefore, the masorah extends to every animal which cross-breeds with chickens.
Turkeys can successfully cross-breed with chickens. Thus:
MW Olsen, US Dept of Agriculture, successfully bred turkey x chicken hybrids from Beltsville Small White turkey hens artificially inseminated with semen from Dark Cornish male chickens. Morphological evidence of the hybrid status of the birds (Olsen, M. W., J. Heredity, 51, 69 (1960)) included the fact that the adult hybrids' dark plumage resembled that of the chicken, being dominant over the recessive white colour of the turkeys. Serological studies on their red cells also provided evidence of the birds being hybrids.
Update, based on Double AA's comment, as to whether this encompasses birds:
Up to this point the universally agreed upon need for a mesorah in order to deem a bird species kosher has been discussed. However, there may be a way around this requirement. The Talmud (Bechorot 7a) mentions a rule known as the "hybridization principle." This principle states that kosher species cannot mate with non-kosher species; hence, the fact that a suspect species can interbreed with a known kosher species confirms the kosher status of the unknown species. In the Talmud it is not explicitly stated if this principle applies only to animals or to birds as well. Many authorities have been willing to rely on the hybridization principle to rule that a bird species is kosher even in the absence of a mesorah. Among them are: Chatam Sofer (Yoreh De'ah 7436 ); Avnei Nezer (Yoreh De'ah 1:75:19-2137 ); Maharsham, (Da'at Torah , Yoreh De'ah 82:338 ); Rav Shmuel Schneerson,39 and Chesed L'Avraham (Tinyana, YD:22-2440 ). All of these authorities seem to view the hybridization principle as a rule separating two disjoined sets. In other words, they are not necessarily subsuming the new species under the mesorah of the known bird. Rather, the known bird is serving to prove that the new species is a member of the set of kosher birds.41
There is a second group of authorities who are willing to accept a weaker form of the hybridization principle.42 Within this group, some accept the stronger form in theory but are unwilling to apply it in practice while others are altogether unsure if it applies to birds. However, they all accept a weaker statement that states that if the unknown species freely chooses to mate with a known kosher species when offered the choice of either its own species or the other species, that is a sufficient indicator to subsume the new unknown species under the mesorah of the known kosher species. This group includes the Netziv (Meshiv Davar , Yoreh De'ah , section 2, 22)43 and Arugat Habosem (Kuntrus Ha'tshuvot , related to siman 82, p. 342-34344 ).
According to Rabbi Michael Broyde (in the article "Is Thanksgiving Kosher?" footnote 15), the question of the permissibility of eating turkey was complicated because there were two birds referred to by the same name. For what is now called a turkey, he quotes Rabbi Shlomoh Kluger who forbids it, and Divrei Chayim, Igros HaBosem (probably the author of Arugas HaBosem quoted in Seth J's answer) and Maharam Shik who permit it. He also references Darkei Tshuvah on both sides. He concludes that the practice in America is to eat it.
The modern bird is so bred to be heavy and fronloaded, that the toms are too clumsy to mate at all, and it is totally dependent (commercial concerns notwithstanding) on AI. If we were to invert the halacha that "what breeds is not triefa" should we be concerned that this bird has been bred into functional sterility? Or that the seven week old market weight chicken is of very low survivability?