The religious way: the only way to become Jewish is if you a born to a Jewish mother. Converts are normally not accepted. And it isn't that hard to find an ex-Soviet Rabbi to write a certificate to prove your Jewish heritage. (Some million, more or less, emphasis on the word less, ex-Soviet Jews in Israel) So if your family is Jewish, you are Jewish.

Catholicism operates on a different basis. Converts are accepted and encouraged, and I think in Catholicism it is passed down by the father.

But, being Jewish has been extended to a quasi-race. The criteria of which is rather arbritary, with the division between Sephardic and Ashkenaic Jews.
The whole thing is still up for debate in Israel.

Do you need to categorize yourself as a Jew? No, save that being Jewish gives you automatic Israeli citizenship, a pretty big incentive.