In Bava Basra 14b, a ברייתא is quoted as saying "Moshe wrote his book (i.e. the Pentateuch), Parashat Bil'am, and [the Book of] Iyov." Why did the ברייתא even raise the issue of Moshe's authorship of Parashat Bil'am?
Does "Parashat Bil'am" in the ברייתא refer to Parashat Balak, which speaks of Bil'am, or another book? If so, do we know more about this book?
R. Menachem Mendel Kasher, in Torah Shelaima (vol. 19 pg. 363) quotes several answers:
Ein Yaakov (Peirush): this is to tell us that despite not being involved in any of the Bilaam story, Moshe was still told this story in all of its details exactly how it was written in the Chumash, and it has a similar status as Sefer Devarim, in that Moshe wasn't told about it before it happened that it would be written down, but only afterwords did God tell him to put it in the Torah. (R. Kasher says that this is also mentioned by the Chosam Sofer in Teshuvah Y.D. 356, where he describes that while the entire Jewish people were witness to all the events described in the Torah from Sefer Shemos onwards, they didn't know about this one, so one might have suspected Moshe of making it up, ch"v)
Rashi there explains that Parshas Balak was singled out because "isn't needed for Moshe or his Torah". The Shelah explains that Rashi means that the story surrounding these prophecies were unnecessary, since Moshe really could have told them all himself being the greatest prophet (and knowing everything known to Bilaam), but because God wanted to put these blessings in the mouth of Yisrael's enemy, the prophecies came about through this story that eventually got placed in the Torah.
Ritva and R. Menachem Tzioni: the 'Parshas Bilaam' that the Gemara is referring to is actually something other than the record of the story that is contained in the Torah. Rather it was something only relevant for its time (perhaps other prophecies that were made by Bilaam), but has since been lost, and hasn't been included in the Tanakh canon at all.
Maharil Diskin in the last page of his teshuvos (and it's quoted in full by Rav Shach in Avi Ezri to Hil. Yesodei HaTorah because he liked it so much) explains that, unlike everything else in Toras Moshe, the words of Bilaam's prophecies were already spoken once to a human being, to Bilaam, which is not true regarding any other prophecy in the Torah (other than those told to Moshe) because only Bilaam and Moshe reached these higher levels of prophecy where God spoke to them with clear words. (An idea similar to this one is found in Nefesh Harav of R. Hershel Schachter)
Nachlas Yaakov (in the Ein Yaakov) says that since the all of the conversations between Bilaam and Balak were in their own foreign language, 'Moshe wrote it' in a way different than the rest of the Torah where he was (al pi Hashem) recording what was happening - here, he was also translating. While that's probably true of many other statements or conversations in the Torah that the words in the Torah are actually a translation (such as when Yosef speaks to his brothers before revealing himself, which the Torah attests was done through a translator), perhaps Parshas Bilaam was singled out because it was an entire parsha written in such a way.
Parashath Bilaam is written with stories that Mosha Rabbeinu never experienced, for example the story with the donkey, no one was there but the donkey bilaam and the maloch. Also the sacrifices and such is from the perspective of bilaam and not Mosha Rabbeinu. Therefore, the gamoro in BB is saying that Mosha Rabbeinu did write it even though the perspective is not from Banei Yisroel but an outside source.
Levi (Louis) Ginsburg in his commentary to the Yerushalmi (p. 289 in this pdf) explains that apparently there were people who claimed that the story of Bilaam was not realy part of the Torah but was added to it from an external source (he brings proof that such a claim existed from other sources in Chazal). Therefore, Chazal wanted to refute this and said that Moshe Rabbeinu wrote this as well and it is indeed part of the Torah.
Maharil Diskind (based on the Rambam in Shemoneh Perakim) explains it like this:
A prophet experiences prophecy through the channel of his personality. Therefore, no two prophets ever prophesize with the same style (Sanhedrin 89a). Every prophet puts the stamp of his/her own personality on their prophecy. However, Moshe Rabbeinu saw the prophecy directly, unadulterated. This is expressed in the gemara's analogy that all other prophets saw through a dim lens, while Moshe saw through a bright lens (Yevamos 49b)- their lenses were tinted with their own personalities, while Moshe saw the pure prophetic image/message.
There was one other exception. Sifri Devarim 357:10:
ולא קם נביא עוד בישראל כמשה בישראל לא קם אבל באומות העולם קם. ואיזה זה? זה בלעם בן בעור
-no prophet rose up among the Jews like Moshe (Devarim 34:10), but among the nations there did arise, and who was it? Bilaam . Bilaam was given the "gift" of pure prophecy as well. (This was because his personality was so filthy that it was impossible to give him prophecy through the conduit of his personality, and in order to give him the prophecy that blessed the Jewish people, it had to circumvent his personal imprint on it.)
Therefore, when Moshe wrote the entire Torah, the prophecies that Adam had experienced, the prophecies that Avraham had experienced, and everyone before him, it was the original presentation of that prophecy in it's purest form. It was his own book because he was the one writing it as a pure expression of the prophecy for the first time. However, the prophecy of Bilaam wasn't Moshe's original work - it had already been prophesized, in it's pure accurate form, by Bilaam.
Therefore, when he wrote down the prophecy of Bilaam, he was just copying it over, restating what had already been stated.
In my shiur on this topic, I present a number of approaches. Some of the ones I had were already mentioned in answers above, so I will skip those, and mention a couple that hadn't been brought up yet:
We would have thought that Bilam wrote it, not Moshe
Rabbeinu Gershom there (BB 14b) writes:
ופרשת בלעם. אע"ג דבלעם גופיה הוה נביא.
And Parshat Bilam – even though Bilam himself was a Navi.
What he seems to be suggesting is that since Bilam was a Navi on the level of Moshe Rabbeinu, we would have thought that Hashem included this section in the Torah just from Bilam himself, who was also a Navi. (Meaning, the Torah would have been G-d's word, co-written by Moshe and Bilam!)
We would have thought that Hashem wrote it, not Moshe
Rabbi Yaakov Kamenetzky (Emet L'Yaakov Devarim 31:11) writes:
ואפשר שזהו גם הפשט במה דאמרינן בב"ב [דף י"ד ע"ב]: משה כתב ספרו ופרשת בלעם וספר איוב, דלכאורה הא פרשת בלעם ג"כ בכלל ספרו היא כמו פרשת קרח וכדומה דבכלל ספרו הן, אלא על כרחך צ"ל דבכל התורה לא היה משה רק כסופר בעלמא - וכלשון הגמרא שם [דף ט"ו ע"א]: הקב"ה אומר ומשה אומר וכותב, אבל משנה תורה הוא עצמו חברו ויסדו, ולפיכך נקרא ספר זה "ספר משה".
And it is possible that this is the simple understanding of what we say in Bava Batra (14b): “Moshe wrote his book and Parshat Bilam and the book of Iyov”, that it seems that Parshat Bilam is included in “his book”, just like Parshat Korach and similar [stories], are part of “his book”. Rather, we are forced to say that for the entire Torah, Moshe only acted as a plain scribe, as the Gemara there (15a) [writes]: “Hashem said, and Moshe [repeated] and wrote”, but Mishneh Torah [Sefer Devarim], he himself [authored], and therefore it is called “Sefer Moshe”.
If this is the case, the Gemara is telling us that Moshe authored Sefer Devarim and the story of Bilam, but everything else in the Torah was just written by him, after G-d composed it. Had the Gemara not mentioned this, we would have thought that G-d himself wrote this section, and dictated it word by word to Moshe. (For those who feel that this is in some way hashkafically inappropriate, see Rav Yosef's comments on Yevamot 4a/Berachot 21b with commentaries there, as well as footnote 3 to this piece in Emet L'Yaakov, which brings similar suggestions from the Gr"a, Ramban, and Or Hachaim.)
It is another book
I will just add one note to the Ritva's suggestion that it was a separate book, which has already been mentioned above - some associate this with the inscription found in Deir Alla, which @DoubleAA mentioned above, and contain the words "Sefer Bilam" or "Book of Bilam". You can read more about it here.
R. Yehoshua ibn Shuaib writes (in his Derashot to Parashat Balak) that it was another book (no longer extant) available in chazal's time (see San 106b) and credited as Moshe's authorship. He says that what appears in parashat Balak was a more resumed portion of that book.
R' Uri Sherqi once said that there are those who say "Parashat Bil`am" referenced in the bera'ita is, in fact, a book external to the Torah.
