Who kicked who out of land, why do two countries claim ownership of the same land, ect
Wow, you don't pick simple questions do you? This isn't my expertise but I'll take a dance with the bear...
So Palestine... The total mess of his region came from several conflicting agendas that smashed into each other in an unholy mess. The region, prior to WW1, was an Ottoman province mostly inhabited by Arabs with a small Jewish population.
During WW1, the British were trying to incite the Arabs to revolt against the Ottoman Turks - in a famous serious of correspondence (the McMahon correspondence), the British foreign secretary gave assurances to Sharif Husayn of Mecca for Arab independence but subject to a few niggling details - the exact borders to be defined later. The British were trying to keep some territory (Lebanon) for French colonial ambitions and Husayn expected there'd be some horse trading but he'd at least get Syria, Palestine, Iraq, Saudi Arabia.
The same year the British and French divided up the Middle-East into spheres of influence (Sykes-Picot Agreement) between the two, to avoid any squabbling and keep their alliance strong. The Brits wanted Palestine (to protect the Suez Canal), Iraq and Iran (their line of communication to India). The French wanted Lebanon and Syria because they wanted a reward for the bloody mess being fought on their land. Many of these points were a violation of the spirit of the earlier promises to Husayn (e.g. Syria and Palestine).
Finally in 1917 the British made the Balfour Declaration of a Jewish national homeland. That came about because the Zionist movement had been lobbying for such a territory as they were sick of the abuse and racism they were facing (and this was pre-WWII). The British also thought that the Jews had undue influence over American and Russian policy (they didn't, but casual racism against Jews worked in their favour for a change) - they were trying to keep the Russians in the war and speed the American entry into the war.
These three competing statements/promises setup the mess that was Palestine. Britain got to administer the 'mandate' and Jews began immigrating. The Arabs (not unnaturally) had no intention to work with the Jews since that would legitimize their new state and to the Arabs, these were foreign intruders on land they'd occupied for centuries. So the Jews began working with the British and setup their own government systems and infrastructure.
Prior to WWII, both the Jews and Arabs realized that the British were their chief impediments. The Arabs revolted and were suppressed (losing men and valuable material). The Jews also turned against the British as they were trying to limit immigration in order to keep the peace with the Arabs. WWII happens and the Jews actually contributed forces to the British which gave them valuable modern training.
Post war, the Brits are trying to restrict Jewish immigration to keep the peace with angry Arabs, the existing Jews started an insurgency and used terrorism to strike back, all the while the Brits were interning Jews fleeing Europe after the Holocaust and looking like colossal dickbags (this is called a PR debacle). Finally the Brits decided this wasn't worth the it and just walked away in 1948.
Now at this time the Jews supported a partition of Palestine (55% of it becoming a Jewish state) and the Arabs opposed this in principle. But the latter had to depend on other Arab states since they'd lost their military forces in the failed revolt of 1938 and those other states had other fish to fry. Trans-Jordan was in fact planning on grabbing some Palestinian land.
1948 came and the British withdrew their forces. Fighting followed as the Jews pushed Arabs out of their projected borders. The Jewish government structures setup earlier transitioned flawlessly into becoming the defacto government. The Palestinian Arabs didn't have a government structure in place and were poorly served by their leaders who all legged it. So on May 15, 1948 Israel is proclaimed which triggered the Arab states to invade. But the Israelis had veteran soldiers from WWII and benefited from interior lines against the various Arab states who attacked piecemeal. Jordan even had a special arrangement with Israel that they only wanted some Palestinian land. So needless to say the Israelis won handily.
This war was the Nakba (catastrophe) for the Palestinians as (the numbers I've been given) 90% became refugees - some fleeing the violence prior to the actual war, some fleeing the war itself. They weren't welcome back and their deserted villages were destroyed and newly immigrated Jews were resettled. So now you've got the origin of that mess. I've tried to be impartial to both sides.
I guess the troubles of Iraq, Iran, Afghanistan and the rest of the Middle East are less prominent today than troubles in Israel but I figured I'd add something about them as well. In your Muslim countries (Iraq, Turkey, Saudi Arabia, etc.) the problems stem mainly (but not completely) from the rough history of early Islam. The violence and minority rule are a mixture of Western civilizations putting certain groups into power as best suit their goals for the countries as other posters have mentioned with Israel. The problem being that each of the different groups of Islam don't see the others as correct and often as lesser peoples. The simple history of these differences is just who the original members of Islam in ~600AD thought was the legitimate heir to Muhammad. Today these different heirs are mainly the basis for the Sunni and Shi'a groups. These groups then spread across the entire Middle East and became the Ottoman Empire and were then split up by the British and French somewhat irregardless of the consequences of putting one group in charge of the other and then mix in the attempts by American, British, French, Soviet, and Israeli forces to exert their own influence on each country and group set up the area much like Serbia and the Baltics at the onset of WWI. I think I've started rambling so I'll sum this all up by saying that many different religions and too much foreign interference generally lead to conflict.
Palestine and Judea, were, in antiquity, ruled by a series of Jewish kingdoms, founded by those who had escaped Egypt.
It was then invaded by Rome in 63 BC and remained a province of Rome, and the Byzantine empire until the recently Islamic Arabs seized the province in around 635 AD.
They had comfortable rulership, allowing Jewish, and Christian pilgrims to travel to important religious sites, the majority of which had a similar significance to them. The Crusades, beginning in 1098 led to 300 years of turbulence as Christians laid claim to what they saw as their promised land.
Israel then became part of the Muslim Ottoman Empire, staying under their rule until 1920. During this time, many Jewish people were forced to leave their countries (see the expulsion of the jews from spain in 1492) settling in the Palestine. The Ottoman empire was welcoming to people of Jewish faith allowing them to settle across their empire.
In 1920 it fell under British and French protection, taken from the Ottomans after WWI. During the 1930s many fled Germany settling in Palestine. This huge influx of people started to put a strain on Arab-Jewish relations as disputes over borders started to arise.
After WWII, the state of Israel was created along with the state of Palestine which included the West Bank, the Gaza strip and the Sinai. The Palestinian Arabs subsequently launched attacks on Israel, promoting the state to retaliate and invade. Whilst some of the areas invaded were taken by Jordan and Egypt, the majority remain under Israeli jurisdiction as occupied territories.
Israel has a complicated history and the problems now are related to events thousands of years ago. It's a difficult situation over there and one which has no easy answer due to the fervent nature of the claims made over ownership.