(1) John's term 'Jew' is limited by geography. So, for instance, Jesus stays in Galilee to avoid the Jews (Jn 7:1), despite that Galilee has a lot of Jews in our sense of the term. The author of the gospel uses the term always and only for Judeans (the one possible exception is probably not really one, see below). Indeed there are a number of groups in the gospel that are not called 'Jews', despite being clearly what we would call Jewish:
(a) the disciples of John
(b) the disciples of Jesus
(c) the Greeks (who are in context clearly Hellenistic Jews, since they have come to Palestine to celebrate the holy days)
(d) Jewish Galileans
In some cases, particularly (a), (b), and perhaps (c), there is a clear opposition of some sort set up between the Jewish group and the Jews in John's sense.
(2) What is more, John usually restricts it even further, to those who are in league with the Pharisees and priests. In other words, the Jews in John's sense of the term are those who are representatives of Judea's religious establishment. Several uses of the term can only be understood in this way. For instance, Caiaphas's advice to the Pharisees and the priests (Jn 11:48ff) is later summarized as his counsel to the Jews (Jn 18:14); and in more than one spot the people of Judea are opposed to the Jews -- the people are afraid of the Jews because (for instance) the Jews have the power to throw them out of the synagogue.
Obviously, there are lots of uses of the term that can be interpreted in more than one way. But there are only two scenes in which the word 'Jew' seems to have a broader signification than this narrower 'those in league with the Pharisees and priests'. The first scene is the discussion between Jesus and the Samaritan woman (Jn 4). The Samaritan woman having called him a Jew, Jesus identifies himself with the Jews:
You Samaritans worship what you do not know; we worship what we do know, for salvation is from the Jews.
If we regarded this use of the term as being closer to our own sense of the term, it would be the only case in the whole gospel which would definitely take that meaning. Much more probable is the conclusion that Jesus is accepting the Samaritan's labeling of him as Judean. We don't find any reason for this in the gospel itself, but, of course, we already know (from Matthew and Luke) that there was a Christian tradition predating this gospel which did, in fact, take Jesus to be born in Judea, despite living in Galilee. Origen had interpreted the phrase 'his own country' in 4:44 (which, if we take it to refer to Galilee seems to conflict with 4:45) to refer to Judea; this is usually dismissed, because modern scholars tend to assume that John only associates Jesus with Galilee (cf. 1:46), not Judea. But I wonder if they are perhaps being too glib on this point, since I suspect Origen is quite right. To identify 'his own country' with Judea fits with the passage preceding. I don't know if this is the right solution or not; but it seems to me that we are taking the gospel more consistently if we regard v. 22 as an affirmation of a Judean identity.
In any case, the other scenario is in some sense more interesting, since it is the interaction between the priests and Pilate (Jn 19). It is a sign of deliberateness, I think, that throughout this entire interaction, the narrator continues to use the term in his usual narrow sense, but Pilate consistently uses it to mean 'Judean'; the Roman governor lumps in all the Judeans together. His title for Jesus, "King of the Jews," clearly means "King of the People of Judea" rather than "King of the Jewish People".
I find all this somewhat interesting in that the gospel is often accused of being anti-Semitic for its heavy, and heavily negative, use of the term 'the Jews'. But the structure of the work tells against this interpretation. There are several Jewish populations who are never condemned under this term, and usually the term is restricted to the Judean religious establishment. The occasional positive use (Jesus with the Samaritan woman) and neutral use (Pilate) are the exceptions that prove the rule: for these are the only cases where there is a straightforward positive argument for taking the term as a reference even to all Judeans. There is no attack on the Jewish people in the Fourth Gospel. Rather, the picture that the evangel gives is of a particular group -- the Judean religious establishment -- who see themselves as the leaders and protectors of all Jewish people, and take this self-appointed mission as a license for meddling in the religious affairs of others. So effective is their meddling that ordinary Jewish people, and even some of their own members, live in fear of their sanction. They are self-appointed gatekeepers, who brand Jewish people who oppose them as sinners or (even worse) Samaritans. That's the picture that John gives; a very limited picture of a very small group within Judaism, and not of the whole Jewish people. In the gospel Jesus opposes not the Jews in general, but the power of the Temple: one might say that in this gospel the Temple of Jesus's body stands over against the corrupted Temple as the symbols of the two opposed ways in which God may be worshipped, the first of which liberates and the second of which oppresses.