Second, the more traditional movements do not always acknowledge the validity of conversions by the more liberal movements.
A more liberal movement might not follow the procedures required by the more traditional movement, thereby invalidating the conversion.
Many people have asked me why traditional Judaism uses matrilineal descent to determine Jewish status, when in all other things (tribal affiliation, priestly status, royalty, etc.) we use patrilineal descent.
In this sense, Judaism is more like a nationality than like other religions, and being Jewish is like a citizenship. This has been established since the earliest days of Judaism.
In the Torah, you will see many references to "the strangers who dwell among you" or "righteous proselytes" or "righteous strangers." These are various classifications of non-Jews who lived among Jews, adopting some or all of the beliefs and practices of Judaism without going through the formal process of conversion and becoming Jews.
In Deuteronomy 7:1-5, in expressing the prohibition against intermarriage, G-d says "he [i.e., the non-Jewish male spouse] will cause your child to turn away from Me and they will worship the gods of others." No such concern is expressed about the child of a non-Jewish female spouse.
From this, we infer that the child of a non-Jewish male spouse is Jewish (and can therefore be turned away from Judaism), but the child of a non-Jewish female spouse is not Jewish (and therefore turning away is not an issue).
However, this technically inaccurate usage is common both within the Jewish community and outside of it, and is therefore used throughout this site.