What's the difference between "Jew", "Israelite" and "Judaizer"? Can these three terms be used interchangeably? Can only one or only two of these terms be applied to one person?
There are several, sometimes overlapping families of terms in play:
This name was given to Abraham's grandson, Jacob, by God and means "God prevails" or "God fights". There's some disagreement over which meaning is accurate, but the source is from this passage in the Torah:
And he said unto him: 'What is thy name?' And he said: 'Jacob.' And he said: 'Thy name shall be called no more Jacob, but Israel; for thou hast striven with God and with men, and hast prevailed.' Genesis 32:28-29 (JPS)
Since Jacob fathered the twelve men who are the founders of the 12 tribes, his name was often used as shorthand for the entire community. We see the same phenomena with Moab, Edom, etc. Sometimes the nation was called Jacob, but Israel is more common. There are several variations:
Israelite—Usually reserved for the descendants of Jacob in the ancient world. While it can be applied to anyone from his own sons on, it's most common for it to mean the nation that formed during the Exodus. When the kingdom divided, the northern portion retained the name until they were defeated by the Assyrians. After that point, the people living in that region were called Samaritans and there is some dispute over the exact relationship with Israel. The "-ite" suffix comes from Greek, which suggests the Septuagint coined the term.
Judah was one of Jacob's sons and the ancestor of David. As such, the tribe of Judah acquired significant political and cultural importance. When the kingdom of Israel split, the southern portion was called Judah even though it included Benjamin and portions of other tribes as well. Like Israel, there are several variations of this name as well:
Jew—Perhaps because of lessons learned from the destruction of the Northern Kingdom, the people of Judah maintained their communal identity while in exile. When the Persian empire conquered the Babylonian empire, the decedents of the kings of Judah where allowed to return to what was now called the province of Yehud. When Alexander conquered Persia, the name became Ιουδαία and under the Romans it became Judea. After a few language jumps, the word arrived in English as Jew. This is the proper term for people who practice the religion described in the Tanakh or have ancestral relationship to Jacob.
Judaizer—This is fairly technical term that comes from one early Christian text which describes the first division of Christianity. Most of the early figures of the Christianity were Jewish. Some hoped to spread the religion to all types of people and others hoped to retain Jewish religious practice. Paul, writing on behalf of the first group, argued that gentiles should not be compelled to adopt Jewish praxis, or "live as do the Jews". The Septuagint used the same verb when translating Esther 8:17:
And in every province, and in every city, whithersoever the king's commandment and his decree came, the Jews had gladness and joy, a feast and a good day. And many from among the peoples of the land became Jews; for the fear of the Jews was fallen upon them.—Esther 8:17 (JPS)
Christian commentators turned the Greek verb into a noun: judaizer. Needless to say, outside of this fairly narrow context, the term is not helpful. It should be avoided, if not because of the possibility of offense, then because it's unlikely to be recognized.
Although not mentioned in the question, "Hebrews" is a commonly used term for the ancestors of Jacob while they were living in Egypt. The name may have derived from Abraham's ancestor, Eber. Other etymologies have been proposed. The word generally signifies the language of the same name, but at various points in history it has been identified with its speakers. The term emphasizes the text, language, and culture of the people rather than their nation or religion.
One of Noah's sons, who is identified by Genesis as an ancestor of Abraham. Many people groups fall under the banner of Semitic, but because the term antisemitism refers to an attitude of hostility to Jews, the word Semite is sometimes mistakenly used for Jews alone.
While there is significant overlap between the term Israelite and Jew, these terms are not always interchangeable. "Judaizer" probably should not be used at all outside of specific domains.
As stated, Israelite is a anglicized version of a word meaning "of the nation of Israel" which referred to people in biblical and post-biblical times. Once the kingdom split after the reign of Solomon, that term which would have applied to the Northern Kingdom of Israel, would not apply to the Southern Kingdom of Judah. The name "Judah" (transliterated better as "Yehudah") is what became "Yehudi"= which we got (through the German I would guess) as "Jew." Of the three it is the only one in use now.
A Judaizer is one who gets others to convert to Judaism or, separately, someone who adopts practices to live in a state which conforms to certain Jewish norms without being a Jew. It is not a term used for Jews or generally, by Jews.
The first two can be used interchangeably, although the use "Israelite" is slightly antiquated now, and is used mostly to refer to the Children of Israel in biblical times. Nowadays, the only term of those three that are used to refer to Jews is, "Jew"
As far as Judaizer, I have come across the term before, but I had to look it up, because it's so uncommon.
The Wikipedia entry on "Judaizer" defines several different uses for the term, none of which would be synonymous with "Jew".
Judaizer is a pejorative term used by Christians to describe those they claim are attempting to change their religion to conform to the laws of Moses. As an example, the Inquisition killed many people who were accused of attempting to "Judaize" the religionas well as being conversos who were accused of trying to (secretly) return to or remain Jewish.
Judaizers is a term for Christians who insist that their co-religionists should follow the Law of Moses. This term is most widely known from its single use in the Greek New Testament (Galatians 2:14) where Paul publicly challenges Peter for compelling gentile converts to Early Christianity to "judaize", also known as the Incident at Antioch.
This term includes groups who claim the necessity of continued obedience to the Law of Moses found in the first five books of the Christian Old Testament. Members of such groups, notably the Seventh-day Adventist Church, dispute the use of the term because "Judaizers" is typically used as a pejorative.
Israelite is the term used up to the end of the second Temple as everyone was a member of the nation which grew fro the descendants of Avraham Yitzchak and Ya'akov (Yisrael or Israel).
Jew After the destruction of the first temple, the term "Jew" became used from the kingdom of Judah, as can be see from the reference in Megillas Esther of "Mordechai Hayehudi (the Jew) who was from the tribe of Benjamin.