You can either be born a Jew, which means that your mother is Jewish, 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, it discourages it. Discouraging conversion helps to filter out those lacking the proper degree of committment. 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. The Torah says it is a special mitzva to love and to be kind to converts even more than to ordinary Jews.