I'll get this out of the way early: my source is Josephus' The Jewish War.

I'm also going to assume that by "initial displacement of Jews" OP means the initiation of the particular displacement of Jews which was supposed to be resolved by the creation of the modern state of Israel.

In around the year 65 CE, the area of Israel was occupied by the Roman Empire. There was a long list of reasons for it: taxation issues, differences in opinion as to what fealty to the Emperor entailed to Jewish subjects, but it's generally agreed that the "sparking incident" was when a man went to the court of the Gentiles in the Temple in Jerusalem, and performed a mock sacrifice of pigeons over a chamber pot. In response to this, a group of Jews, calling themselves Kana'im, meaning "those who are zealous for God" (which is where we get the word Zealot today) tried to organize a war of independence from the Roman Empire. They started by rioting in Jerusalem in 66 in response to the first incident. It went about as well as you'd expect. The Romans came in and had many of them whipped, beaten, and/or crucified. This last one was a big no-no because many of them were Roman Citizens, which made crucifixion illegal (if they got the death sentence, it was supposed to be by beheading). There were further riots in response to this, and eventually a general named Vespasian was dispatched (by Emperor Nero) to deal with it.

Vespasian was generally thought of as even-keeled in terms of his use of force, but he was also very efficient. He kept pushing back the Jewish forces out of northern Judea back into Jerusalem. It didn't hurt that he had 60,000 soldiers with him against basically a lot of much smaller groups at different places and times. Anyway, he was so effective that, in 68 CE, when Nero became so completely insane that nobody in Rome could deal with it anymore, Vespasian was "nominated" as his replacements after Nero's assassination. Unfortunately, so were at least three other people, so Vespasian went off to fight them halfway through the campaign, leaving his son, Titus, to handle the siege of Jerusalem.

Titus, by most accounts, was a dick. Laying siege to Jerusalem was always difficult: it's on a hill, has solid walls, and has its own water supply, so they're not really going anywhere, which means you have to keep pummeling them until there's no infrastructure left. Normally in this position, the tactic is to keep everyone inside until their leaders realize that they're all going to run out of food and negotiate for peace. Alternately, you can let all the people out, leaving a city defended only by what few military people are there, who you can then quickly overrun. Titus did neither of those. He killed anyone who left by crucifixion. This forced as high a body count as possible. Eventually, they kept hammering away at the weakest part of the wall and after 7 months or so Jerusalem fell. Titus was almost immediately made Emperor of Rome after his father was killed (EDIT: died of an illness), so he left the cleanup to another commander, but one of the first things he built was a monument to his victory: The Arch of Titus. It even has a relief of them despoiling the Temple on one side. What Jews remained were either killed in the "cleanup" or were shipped out of Israel as slaves throughout the Roman Empire. Others escaped before the siege of Jerusalem, with some making it as far as Ethiopia and India. Basically, after that, the Temple Judaism system was gone (because there was no way the Romans would allow that thing to be rebuilt), so Rabbis took over as the primary religious teachers and leaders, and because Synagogues could be anywhere, Judaism as a whole entered into a sort of theological and metaphysical holding pattern until such a time as they could return and rebuild the Temple.

TL;DR - Jews in the first century CE started a war of independence against the Roman Empire. It went about as well as you'd expect.

The "initial" displacement of the Jews dates back to the 700s BC when the Assyrians and then the Babylonians invaded Israel and Judah and carried the Hebrews off into slavery.

The first Jewish diaspora, however, took place under the Roman Empire. It began after the First Jewish-Roman War in the late 60s AD, when the Second Temple was destroyed during the Siege of Jerusalem. Jews continued to be persecuted under the Romans for various reasons, including inciting rebellion, over the next century. Finally, under Hadrian, the Bar Kokhbah Rebellion signified the end of major Jewish control and influence in Judea and Jerusalem.

For more information, and interesting reading, I suggest going to wikipedia and reading about the First Jewish-Roman War. Very interesting stuff. For scholarly reading, check out Ruling Class of Judaea by Martin Goodman. Cool stuff.

tl;dr: the romans did it

It's important to note that there are a few different elements in here. First, why Jews migrated outside Israel, and second, why those communities became the dominant jewish communities.

For the first question, the first major movement out was to Babylonia, as other comments mention. In the biblical narrative, some segment of the Jewish establishment is brought to Babylon, and many opt not to return when given the option. This certainly did not affect most Jews, and the extent of it is debatable. Either way, during the second temple period there was a jewish community in Babylonia that was substantial in size, while the temple still stood and there were Jewish political polities in Israel of various sorts.

The second major migration was during the Greek and Roman periods. Israel was controlled variously by various Greek and Roman empires (Ptolemaic, Seleucid, and after a brief period of independence and then Roman domination eventual Roman rule) Jews migrated in significant numbers to other areas, particularly around the Mediterranean. This occurred before the first Roman Jewish war in 70 CE. The Jewish community in the Roman province of Judæa was still the dominant community, but major communities began in more areas. The exact reasons for migration can't be known certainly. It seems some were mercenaries who stayed where they were stationed. Presumably some were moving for economic reasons, as part of Mediterranean trade. But whatever their reasoning, Jews migrated elsewhere, establishing Jewish communities in places like Greece, Rome, and North Africa.

Eventually the Jewish community in Judæa collapsed (or more accurately, fell in size and prestige dramatically over time), leaving the communities outside. Multiple wars with Rome resulted in some forcible migration, but the entire population was not exiled. But the result was also economic decline. The decline of Jewish communities in Israel helped grow communities elsewhere, but those communities aren't just the result if the collapse of Jewish populations in Israel. It's worth noting that the Jewish population didn't disappear entirely. They were simply eclipsed many times over by communities elsewhere.

Jewish communities in the diaspora formed because of migration for economic reasons and when the political situation afforded mobility. Later, war and economic decline caused the diaspora communities become the bulk of the Jewish population.