In a purely technical way "belonging" to Judaism, or in other words, being a Jew, is simple. Your mother needs to be Jewish, as does her mother and her mother before her. Whether by tradition or by proper conversion.
However, I'm sure the answer you're looking for is much more complicated as you used the word "belonging" which is more of a feeling or emotion than actually a concrete fact of being a Jew.
For many religions, simple belief is enough. Not so with Judaism. When a child is born, it doesn't believe in anything, performs no commandments, doesn't keep the Sabbath or Kosher. Yet, the child is still a Jew, like his or her mother. Even more so, even if the mother is non-practicing, she is still Jewish herself, provided her mother was Jewish or that she had a proper conversion.
The reverse is also true. If a non-Jew feels a connection to Judaism and believes in G-d, performs commandments etc.. this does not make them a Jew unless they convert.
Please see the article below for more depth.