Gemara

26a (Mishnah): People in Ma'amados would not fast on Erev Shabbos, due to Kavod Shabbos.

27b: People in Ma'amados would not fast on Erev Shabbos, due to Kavod Shabbos, all the more so on Shabbos itself.

Eruvin 40b - Question (Rabah and other Talmidim of Rav Huna): If a Talmid fasts on Erev Shabbos, should he complete the fast?

Answer (Rava - Beraisa): When Tishah b'Av falls on Erev Shabbos, one may eat k'Beitzah (of food before Shabbos), in order that he not enter Shabbos amidst affliction.

(Beraisa - R. Yehudah): Once, Tishah b'Av fell on Erev Shabbos. They brought a roasted egg for R. Akiva; he ate it without salt. He was not hungry, he just wanted to show the Halachah to Talmidim.

R. Yosi says, he must complete the fast.

R. Yosi: You agree that when Tishah b'Av falls on Motza'ei Shabbos, one must stop eating (on Shabbos) while it is still day. Just like one may end Shabbos amidst (the beginning of) a fast, one may enter Shabbos amidst affliction!

Chachamim: One may end Shabbos amidst a fast, for he ate and drank the entire day (he is satiated). One may not enter amidst affliction of having fasted the entire day!

(Ula): The Halachah follows R. Yosi.

(Mar Zutra citing Rav Huna): The Halachah is, one fasts and completes the fast.

Rishonim

Rif and Rosh (2:25): The Halachah is, one completes a fast on Erev Shabbos.

Rosh (ibid.): This applies even to an individual's fast. The Yerushalmi says that if he did not stipulate to fast only until the Tzibur prays (Ma'ariv), he may fast until Tzeis ha'Kochavim. This is not considered fasting on Shabbos, because a fast that does not last until Shki'ah is not a fast.

Rambam (Hilchos Ta'anis 5:5): Nowadays all of Yisrael have the custom to fast on the 13th of Adar. If the 13th is on Shabbos, we fast on Thursday, the 11th.

Magid Mishnah: The fast cannot be postponed to Sunday, for that is Purim. The Ge'onim say that we fast on Erev Shabbos only if that is the proper day of a Ta'anis, but not if it is pushed off.

Rosh and Ran (Megilah 1:1 and 1b DH Gemara): We fast on the previous Thursday because we normally say many Selichos and prayers on the Ta'anis, so if it was on Erev Shabbos there would not be time to prepare for Shabbos.

Rambam (ibid.): If one of the four fasts fell on Erev Shabbos we fast on that day.

Poskim

Shulchan Aruch (OC 249:3): Anshei Ma'aseh tend to fast every Erev Shabbos.

Beis Yosef (DH v'Yir'eh): In Eruvin, Amora'im asked whether or not a Talmid fasting on Erev Shabbos should complete the fast. The Tur infers that it the way of Anshei Ma'aseh to fast on Erev Shabbos, to enter Shabbos hungry. This is not a proof. However, one may learn from R. Avun, who fasted every Erev Shabbos (Yerushalmi Ta'anis 2:12).

Rebuttal (Bach DH v'Yir'eh and 686 DH ucheshe'Chal): The Tur (686) says that when a fast is pushed off (from Shabbos), we do not fast on Erev Shabbos, for the many Selichos and prayers would not allow time to prepare for Shabbos. However, the Magid Mishnah says that it is due to Kavod Shabbos, i.e. not to enter Shabbos amidst affliction. Erev Shabbos is a little like Shabbos. The conclusion (Eruvin 41b) is that one fasts and completes the fast, i.e. for a fast that should be on Erev Shabbos. One may not elect to fast on Shabbos. Therefore, if one normally fasts on Erev Rosh Chodesh, if Erev Rosh Chodesh is on Shabbos, he fasts on Thursday. The Mordechai and Roke'ach say that an oath to fast on Erev Shabbos is vain; this is from Megilas Ta'anis (Perek 11). In Eruvin we asked about Talmidim who fast on Erev Shabbos. Even though this is improper, the Gemara concluded that they should complete their fast:

Support (Magen Avraham 249:7): The Bach is supported by the Mishnah (26a), Medrash Tanchuma (Bereishis 3), SMaK, Hagahos Minhagim and the She'altos. We must say that R. Avin used to fast every Erev Shabbos because he was an Istnis; if he would eat on Erev Shabbos, he would have no appetite at night.

Note: One could have understand the Mishnah (Ta'anis 26a) like the Tur, that we do not fast on Erev Shabbos lest it detract from Shabbos preparations. However, the Gemara (27b) says that they do not fast on Erev Shabbos due to Kavod Shabbos, all the more so on Shabbos itself. This is like the Bach.

Taz (3): The Tur learns from Eruvin that Talmidim fast on every Erev Shabbos. This does not apply to one who will be weak afterwards and not be able to eat with appetite.

Mishnah Berurah (18): They fast in order to enter Shabbos with appetite. However, the Bach (brought below, 686), Magen Avraham (7) and others say that it is better not to fast on Erev Shabbos to avoid entering Shabbos amidst Inuy, unless he must fast in order to have appetite at night.

Kaf ha'Chayim (21): The fast is in order to scrutinize one's deeds. Through Torah and Teshuvah, Benei Torah merit a great Neshamah Yeserah.

Kaf ha'Chayim (22): One receives Tosefes Neshamah from the fifth hour of Erev Shabbos. Therefore, a fast belittles Kavod Ha'aras Shabbos.

Shulchan Aruch (4): If one accepted to fast on Erev Shabbos he must fast until Tzeis ha'Kochavim, unless he stipulated to fast only until the Tzibur finishes Ma'ariv.

Taz (2): L'Chatchilah one should stipulate so he will not need to fast on part of Shabbos.

Rema: Some say that he should not finish; he should eat immediately after leaving the Beis ha'Keneses. Therefore, one should not complete an individual fast. It is good to stipulate when accepting it.

Shulchan Aruch (OC 550:3): If one of the four fasts fell on Shabbos it is postponed to Sunday.

Rema: If one of the four fasts fell on Erev Shabbos, we read "Va'Ychal..." at Shacharis and Minchah...

Beis Yosef (DH veha'Rav): The Avudraham says that if Asarah b'Teves fell on Shabbos it could not be pushed off, for it says "b'Etzem ha'Yom ha'Zeh", just like it says about Yom Kipur. I do not know his source.

Note: The difficulty is that regarding Yom Kipur it is a proper Drashah, whereas Asarah b'Teves is only mid'Rabanan. Chachamim can enact as they please! Some versions of the Tur cite the Avudraham to say that the other fasts sometimes fall on Shabbos, but Asarah b'Teves does not; if it fell on Shabbos, it could not be pushed off (Hagahos Tur ha'Shalem 4).

Beis Yosef (ibid. and DH Kosav): Of the four fasts, only Asarah b'Teves falls on Erev Shabbos.

Shulchan Aruch (OC 686:2): We fast on the 13th of Adar. If Purim is on Sunday, we fast on the previous Thursday.

Mishnah Berurah (3): If one was away on Thursday and did not know to fast and ate, he should fast on Erev Shabbos.

Rema: If the fast is on Thursday and there is a Bris, it is permitted to eat at the Seudah and to fast on Erev Shabbos.

Rebuttal (Taz 2): We cannot fix the fast for Erev Shabbos, due to Kavod Shabbos! And if we would fast on Erev Shabbos, we must fast until Tzeis ha'Kochavim. If Seudas Shabbos does not override the fast, also Seudas Milah does not! Rather, the Seudah is eaten at night (on Thursday), like for a Bris on Yom Kipur or Tish'ah b'Av. The Anshei Ma'aseh who fast every Erev Shabbos (249:2) enter Shabbos with appetite. For most people, fasting Erev Shabbos will diminish Oneg Shabbos.

Gra (DH v'Im): A Ba'al Bris may eat even on other fasts (except for Yom Kipur and Tish'ah b'Av), even if they are not pushed off.

Mishnah Berurah (7): The Rema permits everyone invited to eat at the Bris, even in the morning, even more than a Minyan. Even the Taz permits the Ba'alei Bris to eat, and they need not compensate. One who is lenient like the Rema will not lose.

Rema (YD 402:12): It is a Mitzvah to fast on the day one's parent died... if it falls on Erev Shabbos, it is like other fasts.