A Jew where that word means "someone who is subject to Halachah" (Jewish law). In this sense of the word "Jew", the convert is an apostate and a heretic. Some people would say that a Christian convert should no longer be called a Jew but an Israelite. Think of this as being like the case of an individual who now resides in a country other than that of his/her birth. His or her birth country might not allow dual nationality, but will restore citizenship to anyone born in the country who wishes to return (in Judaism this is called teshuvah).
A "Jew" where that word means "someone of the Jewish race" according to the "principles" of "scientific racism"; social "race" mythology and the supporters of various looney "biblical race" theories. This is of course all very definitely non-Jewish stuff. Judaism will accept anyone as a convert without prejudice as the "race"; gender or ethnicity and generally treat them no differently than it would a "Jew by birth".
The most common accusation I have met concerns the fact that an adult female convert cannot marry a cohen. This is not because of any doubt concerning the woman's position as a Jew. Cohanim are subject to a number of restrictions (e.g. they can't marry a divorcee either, etc) and one of these is that they can't marry a non-virgin (whether a "Jew by birth" or not). So the question is, without going into distasteful and distressing examinations, how can one be sure that a prospective bride is a virgin? In the case of a girl that has been brought up in the shelter of a Jewish family and is known to reliable witnesses, it is assumed that she is a virgin. In the case of an adult female convert, who knows what she might have got up to before she converted? The best policy is thus thought to be to assumed that she is not a virgin (this saves a lot of needless embarrassment for all concerned). This is not a slight on the individual but just a generalised legal decision aimed a ensuring non-transgression of the Law. However, there is no doubt that she is Jewish.
All of which brings up an interesting further point that illustrates
Judaism's non-racist approach to its converts. Any child of a female convert
to Judaism, who at the time of her mother's conversion is still a minor
in Halachah, is considered to be a "Jew by birth". So a female child who
became Jewish due to her mother's conversion to Judaism can marry
a cohen. This is despite the fact that everyone knows that she was not
a "Jew by birth" in the "race" sense of that term (i.e. she is not what
some Christians refer to as a "genetic Jew"). Isn't Judaism wonderful and
don't some Christians defame it with their racist garbage?