During the last one hundred fifty years or so, the concept of Zionism, the desire of the Jews to run their own state in their historical biblical territory, become prevalent. Theoretically such a state would be free from the periodic persecutions Jews have suffered since the diaspora while living in other people's countries. This resulted in a large number of Jews immigrating to what was the British territory of Palestine in the early 20th century. The persecutions of the era in Europe and Russia were particularly effective in driving this.
The problem of course was that Palestine already had (mostly Muslim) residents. This created a huge bloody mess, culminating in a civil war, which the Jewish militia won, and then a regional war, which the new Jewish armed forces won.
During this whole mess, those who weren't fighting and could get out of Palestine did so (war zones aren't particularly fun for civilians). After the war the victorious Jews allowed Jewish refugees to come back to their homes, but for the most part refused the same courtesy to Muslim Palestinian refugees. Most of them either resettled in the left over bits of former-Palestine that the new country of Israel wasn't claiming, or went into exile in other countries. Many of their descendants are citizens of no country in the world even today (makes passports a problem, no?)
The "leftover bits" of Palestine were mostly claimed by their neighboring Arab countries. However, in the ensuing wars, those same neighboring Arab countries lost all those territories (and often some of their own), so the rest of what used to be Palestine has mostly over the years been stateless areas off-and-on under the domination of the Israeli military.
A couple of those bits, namely what is now called "the west bank of the Jordan", and "The Gaza Strip", many people would like to use to create a new country for Palestinian Arabs. The problem here is that a lot of Jews feel that they won those territories in war fair and square, not to mention the fact that God gave that land to them in the Torah. I don't think its a majority that feel that way, but its an extremely troublesome minority that like to back up their feeling by building Jewish settlements on the land. So what the border between Israel and this purported Palestinian state would be is unknown right now, and would probably require either a treaty or a war to sort out (and Jewish claims keep changing thanks to new settlements).
The Palestinians have been trying hard to build themselves a state in those territories. One big problem there is that Gaza and the west bank aren't physically connected, and after a disputed election now have two competing governments. The government of Gaza, Hamas, doubles as a terrorist organization. The government of the west bank keeps getting its authority undercut by those settlements on what it considers its land, and its inability to negotiate anything with Israel.
So if Palestine is a state, its a state with completely undefined borders, two different competing governments, and without ultimate authority (aka: a monopoly on the use of force) in whatever territory it does possess.
Basically, Palestine isn't a state so much as a bloody mess.