What was the dominant religion of the Arabic people when Prophet Muhammed started the Islamic religion?
In my History of Islam classes there was some review of Pre-Islamic Arabia and a few things were covered, from my class notes we talked about:
Most of this came from class notes and one of our major texts: An Introduction to Islam by Frederick Mathewson Denny
Hope that helps
Pre-Islamic Arabians were polytheistic, worshiping 360 gods; the chief god was the moon-god. They later became monotheistic because of Muhammad.
In 1944, Gertrude Caton-Thompson (1888-1885), an influential English archaeologist, discovered a temple of the moon-god in southern Arabia. The symbols of the crescent moon and no less than 21 inscriptions with the name "sîn" were found in this temple. The temple reveals that it was active even in the Christian era. Evidence gathered from both North and South Arabia demonstrate that moon-god worship was clearly active even in Muhammad's day. (from her book The Tombs and Moon Temple of Hureidah)
The title of this moon-god, sîn, was "al-ilah" which means "the deity" or "the god", meaning he was the chief god among the Arabians' 360 other gods. Its title was commonly used as a name instead of it's actual name.
This god's name was eventually shortened to "allah" in Pre-Islamic times. People back then even used "allah" in naming their children. For example, Muhammad's father's name was Abdullah, meaning "slave of the allah", (they say his uncle's name Abu-Talib has contains the word "allah", but I can't figure it out).
The fact that the word "allah" is never defined in the quran, and that Muhammad assumed that the Arabs already knew who allah was; proves that the pre-Islamic Arabs already worshiped allah. While they believed that allah was the greatest god, Muhammad wanted to go a step further to say that allah was the only god.
For the Arabs, Muhammad said that he still believed in their moon-god allah, but to the Jews and Christians, he said that allah was their god. But, as you can see, allah, al-ilah, sîn, "the moon-god", or whatever you want to call it, is not the God of Abraham, Isaac, and Jacob... יהוה (YHWH or Yahweh)
I take your question as "just before the advent of Muhammad." Therefore I will not delve into ancient history, and limit the answer to late-antiquity and early Middle Ages.
First of all we have to note that people in the Arabic Peninsula could (and can) be divided geographically between the (mostly) arid North (notwithstanding oases and narrow coastal regions) and the prosperous South, with its Sabaean-Yemenite people.
The South was dominated by sedentary kingdoms, and was the Arabia Felix of Roman sources, thanks to prosperous trade and local production of spices. This land became object of a struggle between Jewish and Christan preachers, with alternate phases. Judaism penetrated in the Arabian Peninsula as earlier, and became a crucial region after the destruction of the Temple in Jerusalem in AD 70. Christianity arrived later, but its universalism demanded conversion of the local people. The apex of Judaism was reached with the conversion of the King Dhu Nuwas and the subsequent massacre of Christians of Najraan. Immediately afterwards, with the approval of Byzantium, an expedition of Ethiopians, who were Christians, conquered the Kingdom and brought its independence to an end. Judaism however survived until the advent of Islam.
The other areas of Arabia were instead populated by a complex marriage of sedentary people and nomadic Bedouins. The latter did not leave written records, but we know that the Arabic Hijaz was divided by the Christian zone of influence to the North (centred around Najraan) and the Jewish one to the South (gravitating Yathrib). These areas met in Mecca, which was to become the most important centre in the region, following the fall of the Sabaean South to the Ethiopians and the stalemate in the North between the Sasanid and Byzantine Empires.
The religion in Mecca was complex. Reportedly, at the time of the Quraysh domination, besides Christianity and Judaism a triad of goddesses was venerated: al-Laat, al-Uzzaa and Manaat. These were daughter of Allaah, which was interpreted as either the God of the Bible or another Semitic deity. In these days, Arabia became haeresium ferax, a cradle of heresy.
The importance of Judaism/Christianity was evident in that the people of Mecca, and Arabs in general, considered themselves descendents of Abraham and Ismail, which are Biblical characters.
In the Sacred Months, the Arabs from far and wide would flock to the Holy City of Mecca and its Haram, the sacred precinct, and the Kaaba. They would gather for the Ukaaz, a fair which was also a scene of poetic contests, and climb the Holy Mountain of Arafaat for the pilgrimage. In the sixth century, a new concept of Arabia Sacra was emerging, alongside with the rise of the Arabs, finally aware of being a nation.
Muhammad was born in AD 570 circa.
Source: A Cambridge History of Islam, and references therein.