Aside from the 613 mitzvos that God gave the Jews, He commanded 7 things of all mankind. These are called the “sheva mitzvos b’nei Noach” – the seven laws for the descendants of Noah. Six, or possibly all seven, of these mitzvos were originally commanded to Adam; they were re-stated to Noah after the flood, to be passed down to his descendants, the entire population of the world.
Judaism does not require non-Jews to convert to Judaism. Islam, for example, is a purely monotheistic religion that is perfectly acceptable for Muslims. The Christian doctrine of a Trinity is problematic for Jews but it may be acceptable for Christians. (The concept of worshipping God “with partners” is relevant not only to Christians but also to Hindus. The practical applications of this, however, are far beyond our scope.) Non-Jews are free to relate to God in different ways than Jews. What non-Jews may not do, however, is worship idols in lieu of God.
“Idols” includes not only statues of wood and stone but also the heavenly spheres, works of nature or anything else that one might worship. Even if one acknowledges God as the Creator, he still may not worship an idol under the misguided intention that he is honoring God by honoring one of His servants.
As with the other universal laws, this prohibition contains what are many different mitzvos for Jews: making an idol, having an idol made by others, serving the idol in its normal fashion, etc. Likewise, non-Jews may not engage in the “mystic arts” like divining and necromancy, as these draw a person away from the service of God. Unlike Jews, non-Jews may pretend to bow to an idol in order to save their own lives. (This is what Naaman, the general of Aram, committed to do when his boss, the king, would take him to houses of idolatry. The prophet Elisha approved of this course of action – see 2 Kings chapter 5.)