Before we received the Torah at Sinai, there was no specific legal definition of a Jew, because the Torah had not yet been given. This means that the people who came to Mt. Sinai were not Jews in a “legal” sense. In fact, the Revelation at Mt. Sinai can be viewed as a mass conversion to Judaism of millions of descendants of Abraham. In this sense, every Jew is descended from a convert; some go back to Sinai, and some later in history.
The idea of conversion after the giving of the Torah is mentioned in the Torah - and we are exhorted more than thirty times to treat converts lovingly. One example is, "When a convert lives among you in your land, do not oppress him. The convert shall be like one of your citizens and you shall love him as yourself, for you were strangers in the Land of Egypt" (Lev. 19:33-34).
However, in general we do not encourage conversion. Judaism does not consider those of other religions to be “condemned souls”. A person can be completely righteous and merit the World-to-Come without conversion, by adhering to the basic moral laws revealed to Noah.
The seven Noahide Laws are: no idolatry, blasphemy, murder, theft, intimate immorality, eating from a live animal – and to establish a court system to enforce these laws.
Any non-Jew who keeps these laws in all their detail because God so commanded in the Torah as revealed through Moses, and not simply out of logic, is considered a righteous Gentile deserving the World-to-Come. Nevertheless, any non-Jew who wants to join the Jewish People and observe the Torah in its entirety needs to discuss this in depth with a qualified and experienced Orthodox Rabbi to explore this option.