Which Jew?

If your mother is Jewish then you are Jewish. It doesnt matter whether your mother followed Judaism or not - you are still Jewish, and your children will be Jewish. This means that even if you are for example a reform Jew an Orthodox Jew will recognise you as part of the same family. They may disagree on how your practice your Judaism, but that wont stop them considering you Jewish. The same is true of other branches of Judaism, whether Conservative, or Hasidic.

In theory someone converting to Judaism - is not really a convert, instead they are someone who has a Jewish soul which has found its way home. The problem is that Orthodox rabbis do not accept reform convertions. This means that while the convert and the reform movement may accept them and all future generaitons as converts, the rest of judaism will reject them and their decendents on the basis that the convertion was not valid.

This is a very heavy burden to place on future generations - that they will only be accepted fully within the jewish community if they recive an orthodox convertion and that until someone does so all generations will be in this strange middle world where they are conisdered Jewish by some - but not by others.