You sure you want to open up this can of worms? :-)
Here's the situation. There is no explicit mention of any such concept in the Torah, Talmud, or adressed by the Rambam, the Rosh, the Tur, or the Shulchan Aruch. The first time this really became an issue when during WWII when yeshiva students (notably those from Mir and Chachmei Lublin) relocated from Europe to Japan. They contacted various rabbis in Israel asking them where the dateline was, because they needed to know when to celebrate shabbat and various holidays.
They got two responses:
- The Chazon Ish based his opinion on a comment of the Baal Ha'Meor (Zerachiah ha-Levi of Girona). The Baal Ha'Meor had a story about kidush hachodesh, which didn't address the dateline directly, but did maintain that the day "started" 90 degrees east of Jerusalem. The Chazon Ish therefore calculated that the dateline is 90* east of Jlem, (about the 125th E meridian), but it also bends around land masses. This means that since it hits in the middle of the Asian land mass, it would bend and go along the coast of China. China and mainland Asia would be in the same halachic day (same side of the dateline) as the secular calendar would have you believe. The islands (Japan, the Philippines, etc.) would actually be on the other side of the dateline and would NOT be in the same day as the regular calendar would have you believe. In other words, according to the Chazon Ish, Sunday is shabbat in Japan.
- The second answer was from Yechiel Michel Tuchatzinsky, who maintained that since Jerusalem is the center of the world, the dateline must be 180* (exactly across from) Jerusalem, which would put it at about 144.8 W meridian. This is also the opinion of Yehuda Henkin, Moshe Feinstein, and Shlomo Zalman Auerbach, and Yosef Eliashiv.
Since that initial disagreement, there have been a few other opinions:
- Sefer ha'ibur - The center of the world is 23.5* or 24* east of Jlem, and then using the proof of the Baal Ha'meor, the dateline would be 90* east of that.
- Rav Isser Zalman Meltzer and Rav Tzvi Pesach Frank - there is no halachic dateline at all. It isn't mentioned in the talmud, tur, or shulchan aruch. Therefore people should just follow the secular international dateline.
- Rav Yonatan Shteif - the date is determined by tradition. If the day is traditionally Saturday, then that's the day shabbat is. The difference between this opinion and the one above is that if the dateline moved, Rav Shteif would say the halachic dateline would not.
- Rav David Shapiro - the line is 135* East of Jlem.
- Yonah Merzbach - take the easternmost point of Asia (the tip of Sibera by the Bering straight), and use that longitudinal line as the halachic dateline. This is about 170W
In terms of what happened historically, there are reports of yeshivas in Japan during the WWII period who kept shabbat on Sunday, although to avoid maarit ayin (looking like they were violating shabbat), they also refrained from melacha on Saturday, even if they did the weekday davening and put on tefilin. However this is a historic anomaly. Jewish communities in the affected areas (Japan, Hawaii, etc.) have always kept shabbat on Saturday, essentially following the opinion of Rav Isser Zalman Meltzer and Tzvi Pesach Frank.
That enough for you? If you want more, I suggest "the dateline in halacha" by Zalman Tropper in English, or תאריך ישראל in Hebrew.