Why do we say Hallel on Rosh Chodesh? On the festivals we are thanking G-D for a salvation that occurred in the past on that date, but Rosh Chodesh does not commemorate any specific event that took place, so what are we thanking for?
Hallel on Rosh Chodesh says that this is actually a minhag that has been acceptedby Bnai Yisrael to show that the date has kedusha even though it is not required because we are allowed to do melacha. There is a case in the talmud (from memory) in which an amora was going to stop the Baal Tefilah from saying Hallel until he saw that part of it was left out. Thanks to @RonP for pointing out that this story is about Rav in Ta'anis 28b.
There is a widespread custom to recite Hallel on Rosh Chodesh. Indeed, according to the letter of the law, there is no obligation to do so, because Hallel is required only on days that are called mo’ed (appointed times), and when work is prohibited. Rosh Chodesh is called a mo’ed, but work is permitted on the day. Still, the Jewish people have accepted the practice of saying Hallel on Rosh Chodesh, in order to give expression to the sanctity of the day, which is capable of elevating one to the level of singing God’s praises. To make it clear that the Hallel of Rosh Chodesh is based on custom, and not obligatory, we omit two paragraphs that are said when reciting the complete Hallel (the complete Hallel consists of chapters 113-118 of Tehillim, and we skip 115:1-11 and 116:1-11).
The issue of Hallel on Rosh Chodesh is elucidated in Arachin 10b, Ta’anit 28b, Tosafot ibid., and Tosafot Berachot 14a. See Sefer Rosh Chodesh, chap. 6, sections 1-6 and 20, with notes. Notes 2 and 27, there, explain the reason for the custom. There is another type of obligatory Hallel, which is not dependent on the sanctity of a day, rather on a salvation, like the one said on Chanukah. See below 4:6 and 11:8.
In fact, there is also a machlokes as to whether or not to say the bracha on Hallel. Follow the link to see more about that.
The answer above is very good and provide a very valuable introduction for the Sugiah in Gemoro. (Tosfoth Berachoth 14A is the main source "al Hadaf".)
But I dare allow myself an additional response because the OP has not been treated.
With the help of the Taamey Minhagim 436 I found the answer in the Tanya Rabbati (right column,second paragraph, I typed it below)
מה שאנו קורין הלל בראש חודש מצאתי בשם הגאונים ז"ל מניין שאומרים הלל בראשי חודשים מצינו שרמזו דוד תהילים ק"נ הללויה הללו אל בקדשו י"ב פעמים הללו כנגד י"ב חודשים ומשמע הללו אל בקדשו על קידוש החודש הללויה על שנה מעוברת י"ג חודשים ועל חודש שני ימים
In English. The Geonim relate the last psalm of David (number 150). It includes any instigation to praise G_d. It appears 12 times, which refers to the 12 months of the year (and to justify the years of 13 months and the fact that sometimes there are two days of new moon, a verse adds praise G_d when you sanctify the new moon) .
 I still want to clarify, even though it is evoked in the debate of the Gemara (Arachin 10B) that Rosh Chodesh is a Moed, this is not retained. The Gemara looking for a reason to make Rosh Chodesh an equivalent of Moadim (Holidays (I dont want speak here on the difference between 7 of Pessach and others holidays)). She wanted to suggest the Hallel there is a mitzvah. But this attempt failed, since reading the Hallel of Rosh Chodesh is a minhag. So the reason of the minhag is not discussed.
 Tanya Rabbati A Rishon from Italy.