There are two ways someone can be a Jew. You can either be born a Jew or you can convert. (A convert is called a ger which literally means stranger.)
Being born a Jew is pretty simple. If your mother is Jewish then so are you, if she isn’t then neither are you. It doesn’t matter whether your father is Jewish or not.
Conversion is much more complicated. Judaism does not actively encourage conversion, in fact, to a large degree it actually discourages it. This is the reason Jews have never had missionaries trying to convert non-Jews. This does not mean, however, that Judaism doesn’t want the convert. Judaism does want the convert but he must be 100% committed. Discouraging conversion helps to filter out those lacking the proper degree of commitment.
Conversion requires a beis din (a Jewish court of law) made up of three men qualified to serve on such a court. When a non-Jew wants to convert, the court is required to investigate the potential convert and make sure he or she has no ulterior motives (such as wanting to marry a Jew). If any such intent is found the court may not accept him or her for conversion. But if we see that the non-Jew sincerely wants to convert then the court is required to ask him, "Why do you want to become a Jew? Don’t you see that Jews are oppressed throughout the world?". If he leaves, fine. If he still wants to become a Jew the court must then inform him of the mitzvos (commandments) that a Jew must keep. We don’t have to tell him every mitzva but we must give examples of some of the hard mitzvos and some of the easy ones. We also tell him about the rewards that come from keeping the mitzvos and the punishments that come from not keeping them. We tell him that as a non-Jew he is not required to keep the mitzvos and he is not punished if he violates them but he also isn’t rewarded for keeping them, as a Jew he will be rewarded for keeping the mitzvos but he will also be punished for not keeping them. We teach him that the only true righteousness is knowing and keeping the mitzvos. We teach him about the history of the Jews, that, no matter what happens, the Jews will never be destroyed, unlike all the other nations of the world which only last for a short time and then they disappear.
If he still wants to be a Jew, we circumcise him and, after he is healed, he immerses himself in a mikva. (A mikva is a special pool of water which is used for many religious purposes in Judaism. It must be made according to very specific rules.) A female convert only has to immerse herself.
Once someone has converted to Judaism they have the full status of Jews. They are Jews in every way, and, just like any other Jew, they can never cease to be Jews.
There is a special mitzva to love and to be kind to converts even more than ordinary Jews, like it says in the Torah, Devarim 10:19, "You shall love the convert, for you were strangers in the land of Egypt." (This is saying that, since we were once strangers in Egypt, we should understand how difficult it is for someone to adjust to a new world and we should be kind to him.) The Torah also has a special mitzva not to be unkind to a convert in Shemos [Exodus] 22:20, "Do not hurt the feelings of a convert or oppress him, for you were strangers in the land of Egypt." and also in Shemos 23:9, "Do not oppress a convert, you know how it feels to be a stranger because you were strangers in the land of Egypt." And in VaYikra [Leviticus] 19:33-34, "When a convert comes to live in your land, do not hurt his feelings. The convert should be to you exactly like a born Jew and you shall love him like yourself, for you were strangers in Egypt - I am HaShem, your God."
