The Gemara in Masechet Shabbat establishes the obligation to add Al Ha'nisim – together with the paragraph of "Bi'mei Matityahu" – in the Amida prayer during Chanukah. We insert Al Ha'nisim in the section of Modim, which expresses our gratitude to the Almighty and is thus an appropriate context for the recitation of Al Ha'nisim, through which we give praise to G-d for the great miracle of Chanukah.
The question is asked as to whether or not we should begin reciting Al Ha'nisim already during Arbit on the first night of Chanukah. Since we recite Arbit on Chanukah before lighting the candles, perhaps we should not yet mention the miracle of Chanukah in the Arbit service on the first night, before we have lit candles. Indeed, there are views among the Rishonim and Geonim that on Purim, which also requires the recitation of Al Ha'nisim, the Al Ha'nisim prayer is not recited during Arbit, since the Megila has not yet been read. Correspondingly, one might argue that we should not recite Al Ha'nisim in Arbit on the first night of Chanukah, before we have lit the candles to commemorate the miracle.
In truth, however, Halacha requires reciting Al Ha'nisim in Arbit even on the first night of Chanukah, despite the fact that one has yet to light the Chanukah candles. And although Rav Amram Gaon maintains that Al Ha'nisim is not recited on the first night of Chanukah, Halacha does not follow his opinion. In fact, Chacham Ovadia Yoseph, in "Kol Sinai," argues that even those who hold that Al Ha'nisim is not recited on the night of Purim, as mentioned above, would agree that one does recite Al Ha'nisim on the first night of Chanukah. Chanukah, he explains, commemorates two miracles: the miracle of the oil, and the miracle of the Jews' military victory over the Greeks. Therefore, even before we light the Chanukah candles on the first night, which begins our commemoration of the miracle of the oil, we must still celebrate and give praise for the other miracle – the military triumph – which occurred immediately. Hence, even according to the view that on Purim we should begin reciting Al Ha'nisim only in the morning, on Chanukah we begin immediately at Arbit.
It should be noted that even if one recites Arbit before sundown, after the point of Pelag Ha'mincha (the earliest time one may recite the Arbit service), he includes Al Ha'nisim in the Amida prayer. (Of course, it is not very common in the wintertime to recite Arbit before sundown.)
Another question that was addressed concerning the recitation of Al Ha'nisim on Chanukah is whether one recites it in the Musaf prayer on Shabbat of Chanukah, and on Rosh Chodesh Tevet (which always occurs during Chanukah). Since Musaf is required not due to the festival of Chanukah, but rather because of the occasion of Shabbat or Rosh Chodesh, one might have argued against including Al Ha'nissim in the Musaf prayer. The Gemara addresses this issue and concludes that one does, in fact, insert Al Ha'nisim in the Musaf prayer during Chanukah. Since that day obligates one in the recitation of Musaf, and that day is Chanukah, all the prayers of that day – including Musaf – require the inclusion of Al Ha'nissim.
Summary: One adds Al Ha'nissim in the Modim section of every Amida prayer recited on Chanukah, including the Arbit service on the first night, and including the Musaf prayer on Shabbat and on Rosh Chodesh Tevet.