Naturally there is a debate over this question and therefore there are different answers.
The question of who owns Palestine does not have a simple answer. The struggle between Israelis and Palestinians goes back to ancient times, when the ancient Israelites lived in and around Palestine and fought many wars with their neighbors. In a sense, the current conflict is an extension of those religious wars. But the modern conflict has its roots in modern times, specifically in the disintegration of a vast empire.
The land called Palestine was occupied by the Ottoman Empire, a large group of territories ruled by an oppressive regime that found itself on the losing end of World War I. Great Britain had the most troops in Palestine when the war ended, and so Britain "won" the right to administer Palestine. Other territories became independent; Palestine did not.
One of the many actions taken by the British government and army was the announcement and enforcement of the Balfour Declaration, which stated that the British people and soldiers supported the construction of a homeland for the Jewish people in Palestine. This was way back in 1917, long before the Holocaust
Beginning in 1922, large numbers of Jewish people migrated to Palestine, pursuant to the Balfour Declaration. This migration continued for the rest of the decade and accelerated in the 1930s and 1940s. The people who called Palestine their homeland at this time didn't take too well to large numbers of new people moving in, especially since those "new neighbors" were Jewish and the majority of the people who were living in Palestine at the time were Muslim. In 1937, many Palestinians rebelled, calling for an independent nation, just like their neighbors were granted. Great Britain tried to find a way to satisfy both sides but gave up and, after the end of World War II, turned the problem over to the newly formed United Nations
The U.N. proposed side-by-side Israeli and Palestinian states, with Jerusalem being part of both. Jews flocked to the area by the thousands after the Holocaust. Israel proclaimed its independence in 1948 and promptly set about occupying three-quarters of the Palestinian state, including part of Jerusalem. Jordan and Egypt occupied the other part, and most of the Palestinians fled for their lives.
Tensions flared between the neighboring nations for years. In 1967, Israel struck out against Egypt and Jordan. This was the Six-Day War, and it resulted in the expansion of Israel into all of Palestinian territory and land formerly claimed by Egypt and Jordan, including the Sinai Peninsula, the West Bank, the Gaza Strip, the Golan Heights, and (of course) all of Jerusalem.
The United Nations called on Israel to give back the territory it had seized, but the calls fell on deaf ears. Israel controls this territory to this day. After this, Israel continued to insist that it had earned the right to occupy these territories. Palestinians, on the other hand, expressed what they saw as their right to live in a land that the U.N. recognized as theirs.
Wars of words led to wars with bullets and tanks. Both sides had "hawks" who thought that violence was the answer. One of the main hawks on the Palestinian side was Yasser Arafat, leader of the Palestine Liberation Organization and of Fatah, a guerilla movement that had violence as its goal. Members of Fatah were responsible for the slaughter of Israeli athletes at the Olympic Games in Munich in 1972.
But Israelis weren't innocent, either. Israeli leaders tended to make statements and take actions that inflamed the situation. Israeli Prime Ministers refused to even address Arafat as the Palestinian leader. Arafat refused to call off the hijackings, bombings, and kidnappings.
An increase in Israeli attacks on Palestinian leaders and territory led to the declaration of an intifada ("uprising") in 1987. It lasted six years, and it led Palestinian people to question Israeli people and methods with guns and bombs and widespread distrust. Israel, of course, responded with even more determination to keep the upper hand. Often, Israeli "demonstrations of force" aimed at keeping Palestinian guerrillas in line led to civilian casualties.
Arafat, meanwhile, was trying to cement his role as a statesman, even if he wasn't the leader of a state. He addressed the U.N. again, in 1988, and renounced terrorism as a means to reclaiming land for his people. Some observers found this declaration to be quite meaningful; others dismissed it as rhetoric not to be trusted.
With the ascension of Yitzhak Rabin to the post of Prime Minister of Israel came a new era in Mideast peace talks. Rabin and Arafat negotiated in secret, resulting in the historic Oslo Accord, which gave the Palestinians living in Israel-occupied territory much more of the say in their daily affairs and also recognized Arafat as a partner in the peace process. (He had formerly been labeled a "terrorist" and someone who couldn't be trusted.) Arafat and Rabin stood on the same stage at the American White House and shook hands, with the American President, Bill Clinton, looking on, on Sept. 13, 1993. The very next year, Arafat and Rabin shared the Nobel Peace Prize with Israeli Foreign Minister Shimon Peres. Also in 1994, Arafat returned to the Gaza Strip after 26 years in exile.
Things looked up for the Palestinian people for a while in the late 1990s, despite the assassination of Rabin in 1995. Arafat and the Israeli government signed an agreement that provided for the removal of Israeli settlers and soldiers from most of the West Bank city of Hebron in 1997. And in 1998, another agreement was signed by both sides, furthering the peaceful settlement of the West Bank "problem."
But with the coming of the new millenium, the peace process unraveled again. Frustrated by the lack of real progress, Arafat and the PLO declared a second intifada. It is still in effect. In response, Israel moved tanks and soldiers into position around Arafat's headquarters in Ramallah, effectively putting him under siege. He was allowed to leave only when he was too ill to survive in West Bank hospitals.
For Israel, the story hasn't changed in many years. Israelis have settled in to lands that Palestinians call home, and these Israelis believe that they have the right to live there and call it home themselves. Palestinians want the Israeli "occupiers" to leave, and they desperately want a homeland of their own, a country of their own. That is something that Yasser Arafat, in the end, could not deliver. Perhaps his death will be the first step in breaking the polarization that has so gripped both sides in the past 20 years.
Palestine is part of the Levant, which is the area east of the Mediterranean where Syria, Lebanon, Jordan and modern day Israel are. The Palestinians are mostly Arabs, Muslims and Christians, with some other ethnic groups who lived in Palestine for thousands of years. The struggle between Jews and Palestinians goes back to the late 1800's when the Zionist movement decided to establish a home for the Jews in Palestine.
At the turn of the 20th century, the Levant including Palestine was part of the Ottoman Empire. The Arabs with the help of the British rebelled ageist the Ottomans in hope of gaining independence, however; soon after the Turkish forces retreated, the French and the British forces moved in to occupy Syria, Lebanon, Jordan and Palestine in a system called "Mandate" which is another word for Colonialism.
The U.N. later proposed side-by-side Israeli and Palestinian states, with Jerusalem being an international territory. The U.N. resolution gave 60% of Palestine to the Jews and 40% to the Palestinians even though the Palestinians accounted for 60% of the population at the time. In 1974 the Jews stepped up their campaign against the Palestinians, spreading fear and terror in Palestine. In one day, 265 Palestinians were slaughtered in the village of Deir Yassin. Israel proclaimed its independence in 1948 and promptly set about occupying three-quarters of the Palestinian state, including part of Jerusalem killing 17,000 Palestinians. Jordan and Egypt occupied the other part, and more than one million Palestinians became refugees in neighboring countries, the West Bank and Gaza Strip.
When the world failed to return the Palestinians to their homes and land, they resolved to armed struggle against Israel led by Yasser Arafat, leader of the Palestine Liberation Organization and of Fatah, a guerrilla movement that had the liberation of Palestine as its goal.
An increase in Israeli attacks on Palestinian leaders inside and outside Palestine and the continuation of the occupation to the West Bank and Gaza led to the declaration of an intifada ("uprising") in 1987. It lasted six years. Palestinians continues to confront the Israeli occupation. There was a peace process, resulting in the partial liberation of Gaza, but this was not to last.
But with the coming of the new millennium the peace process unraveled again. Frustrated by the lack of real progress, the continuous building of Jewish settlements in the West Bank and the continuous incursions of the Israeli forces into the Palestinian territories, the Palestinians went into a second Intifadah (uprising). It is still in effect. In response, Israel moved tanks and soldiers into position around Arafat's headquarters in Ramallah, effectively putting him under siege. He was allowed to leave only when he was too ill to survive in West Bank hospitals.
The story continues for the Palestinians. In 2009, they are still under a occupation surrounded by walls and having to go through 100's of Israeli checkpoint and economic blockage. The situation is much worse in the Gaza Strip, where 1.5 million Palestinians have been leaving under siege for the last 18 months. In December 2008, Israel launched an attack against Hamas killing 1300 and destroying the infrastructure of Gaza.
The ultimate goal of Israel is to make life so miserable for the Palestinians that they would pick up and leave the land and move to any number of other Muslim states available to them throughout the world.
Framing the question as who owns Palestine already sets the question as having a certain desired answer and also the lack of definitions for what constitutes "ownership" and what constitutes "Palestine" make the question difficult to answer.
As for "Palestine" this term is typically interpreted one of two ways. The first way is to refer to all of the land in the British Mandate of Palestine which includes the Modern State of Israel (except for the Golan Heights), the Gaza Strip, and the West Bank. The second way is to refer to exclusively those territories which the Palestinian Authority claims will serve as a basis for a future Palestinian State: the West Bank and Gaza Strip. Understandably, it changes the argument fundamentally if 78% of the territory in question is exempted from the discussion.
As for "ownership" there are two understandings of this word. The first is the literal definition of possessing something. Therefore whatever persons, companies, organizations, or governments own something with proper title as viewed by recognized governments are those who have "ownership". (This is like any typical sale.) The second definition is the perceived Color of Right of Title, which is to say that a certain person, organization, or government should have proper title but does not have it on account of an illegal activity. (This case would come for example if A owned a book and B stole it. While B has physical possession of the book, A still retains ownership since stealing, the act of transfer and acquisition, is illegal.) Understandably, most Israelis claim that no illegal act took place and therefore title properly belongs to them. Palestinians and their sympathizers often (but not always) argue that their land was stolen and therefore, they retain proper ownership.
As for the 78% of Mandatory Palestine which is now the State of Israel, this came about through Israel's acceptance of UN Resolution 181 and its border defense against Arab aggression to counter international laws that they did not like. As a result, the acquisition in the 1947-1949 of war was not an illegal act since self-defense is not a criminal act unless it is grossly disproportionate to the attack and the war was a relatively balanced affair as well as being resolved at the moment that each Arab state was willing to engage in an armistice. Just to clarify, this means that the 1949 borders of Israel belong to Israel.
As for the remaining Palestinian areas that were acquired in 1967, the situation becomes murkier, but as concerns the West Bank, Jordan attacked Israel first and Israel retaliated. Again the self-defense doctrine comes to the fore. Israel would have the rights to those territories acquired in self-defense. However, Israel was willing to concede some of those rights pursuant to a final, lasting peace with its neighbors, which is all that UN Resolution 242 discusses. Those states that have made peace with Israel since 1967 (Egypt and Jordan) have seen the return of territory (in Egypt's case) or the cession of the rights to occupied territory with tertiary partners (Jordan to the Palestinian Authority).
As for Gaza specifically, at this point in history, there are only two parties that claim it, the Palestinian Authority and Hamas. Since Hamas is not recognized as an independent government by any country, it retains exclusively de facto control of the area while the Palestinian Authority retains de jurecontrol pursuant to the Oslo Accords of 1993.
As for the West Bank specifically, since Israel ceded the possible claim to the West Bank to Jordan in UN Resolution 242 and Jordan then agreed to cede its rights to claim the land in full to the Palestinian Authority, the Palestinian Authority has the right to claim the lands of the West Bank. There are certainly Israelis who disagree with the extent of these rights, their viability vis à vis their religion, or use the doctrine of terra nullius to disavow the Palestinian Authority of these rights, but this is the minority of Israelis. Most Israelis want the majority of the West Bank to revert to the Palestinian Authority with a guarantee of peaceful coexistence.