“”Oh, the Protestants hate the Catholics,
And the Catholics hate the Protestants,
|—Tom Lehrer, National Brotherhood Week|
While the term anti-Semitism has its roots in prejudice against the so-called Semitic peoples—the Arabs and Jews of the Levant—over the course of the twentieth century it came to refer exclusively to anti-Jewish attitudes and actions, which have taken a number of forms along a spectrum that ranges from discourses that paint Jews as embodying particular stereotypical characteristics to genocide.
 General prejudice
Most such beliefs feed off of ignorance and prejudice that arises from historical interpretations of Christianity and its teachings, and have traditionally been fanned with accusations of heresy and obvious cultural differences, as well as Biblical associations of Jewish leaders (often referred to simply as "the Jews", or that perennial sign of Nazi beliefs, "the Jew", by the ignorant and those who tend to misspeak) with the death of Jesus.
Anti-Semites frequently claim that Jews secretly control banks/governments/the media/the Treaty of Versailles. It is similar to racism in that it frequently applies a stereotype or stereotypes to all members of a diverse group. Nazism took anti-Semitism to extremes, resulting in the Holocaust. Modern anti-Semitism frequently takes the form of Holocaust denial or mass media/governmental conspiracy theories such as the "Zionist Occupational Government" beloved of white supremacists, but classic anti-semitic tracts including the infamous 19th c. Russian forgery The Protocols of the Learned Elders of Zion have become increasingly popular in the Middle East. Some lunatic anti-semite may think almost every person he/she doesn't like (or is rich) is Jewish, even when they aren't. .
Both the left and right are guilty of antisemitism. Ruth Fischer, a Communist leader of Weimar-era Germany, called for "Jewish capitalists" to be hanged from lampposts; ironically her contemporary Hitler, like many other right-wing antisemites, considered Communism to be a Jewish plot.
 Zionism and anti-Zionism
Anti-zionism is not antisemitism, however, some antisemites use anti-Zionism as a kind of cover and entry-level recruiting tool. In addition, most antisemites see Zionism not as a modern movement to establish a Jewish state in Palestine, but as some kind of ancient, all-encompassing world conspiracy — in their language, the term "Zionism" means more or less the same as "The Jews™" It's embarrassing for advocates of the Palestinian cause.
On the flip side, Israel's staunchest gentile defenders in the United States tend to be extreme evangelical Protestants, who eagerly look forward to the "ingathering" of Jews in Israel followed by their massacre and/or forced conversion to Christianity. John Hagee, one of these tireless soldiers for Christ and Israel, got into a little trouble after opining that the Nazi Holocaust was all part of God's plan to punish European Jews for being too irreligious, or something. Strange bedfellows.
 Khazar myth
Some anti-Semites attempt to discredit Israel and the ethnicity of many Jews as a whole by claiming that the Ashkenazi Jews (i.e., those descended from a German bloodline) are not actually Jews, but descended from the Turkic Khazar Empire. This myth has no evidence behind it -- while the Khazar ruling class did adopt Judaism and Jews settled in the Empire, they had already inhabited Europe long before it came about and the historical and genetic evidence doesn't fit this pseudohistorical "theory" either. This is often used as a way to justify the elimination of Israel and/or to explain why the British/Aryan/insert race here are the REAL chosen people of the bible, and not the Jews.