Main Issues
In 1967, Israel fought the Six-Day War against the Arab States and took over control of the West Bank and Gaza. These territories did not come under Israeli Civil Authority and have been instead militarily controlled. Palestinians who live in these territories have to contend with Israeli checkpoints, military provisions, and incoming settlers. The Settlers cause a number of problems (see below). This occupation is perceived by Palestinians to be a repression of their Right to a State and their ability to lead normal productive lives.
Due to the militancy of Hamas, Israel has found it necessary for defense to form a blockade around Gaza and to only allow certain materials into the territory. This has resulted in a Human Relief Crisis in the Gaza where the average caught in the struggle barely have enough food, heat, and light to adequately survive. On a different token, the West Bank (as controlled by the Palestine Authority) is a patchwork of separate unconnected jurisdictions. As a result, the West Bank leaders depend on Israel for defense coordination, tax collection, and assurances of safety from settlers. This creates a secondary occupation-dynamic where the Palestinian government is bound to the desires and wishes of the Israeli people in addition to its actual constituency.
More on Settlers
Close to 500,000 Israelis live in the West Bank or East Jerusalem, the Palestinian Territories, in several major cities like Ariel, Modiin Illit, East Jerusalem, Pisgat Zeev, Ma'ale Adumim, and Efrat. Entire neighborhoods, schools, universities, companies, and livelihoods have been built up in the West Bank. In order to build to these settlements, land in the West Bank is confiscated according to a variety of arcane, inane, and relatively unjust laws (depending on your perspective). In addition to the lands taken in order for the Settlers to live, Settlers have also taken sufficient lands to build highways and other infrastructure to connect the Settlements to 1949 Israel, bypassing Palestinian cities and making it more difficult for Palestinians to move between different Palestinian cities in the territories. Several Israeli Settler Highways even prohibit Palestinian usage.
The Settlers believe that they have the right to live closer to the various holy sites in the West Bank and East Jerusalem. For many of them, this is part of why God allowed Israel to conquer the West Bank in the first place. They want to raise their children and build a life near where Jewish civilization actually began, in the hills of Judea and Samaria which form the West Bank.
Finally, the Settlers live under Israeli Civil Law even though their Palestinian Neighbors live under Israeli Martial Law. This creates a weird dynamic where Settlers who commit crimes are judged in 1949 Israel in front a judge and Israeli jury. Palestinians who commit crimes are judged in the West Bank at a military tribunal presided over by an Israeli Army Judge. It creates incentives for Israeli Settler recklessness since Palestinians are likelier to face worse treatment for counterattacks than the Settlers for the original attacks. This is not to say that Palestinians are innocent in the bloody Settler-Palestinian clashes, but that the legal structure makes it easier for Israeli Settlers to provoke Palestinians without worrying about the consequences.
Palestinians have limited control over certain areas, primarily the Gaza Strip and parts of the West Bank, where the Palestinian Authority and Hamas operate. However, their control is often restricted by Israeli military presence and settlements, leading to a complex political and territorial situation. In Gaza, Hamas governs, while the West Bank is administratively divided into areas with varying degrees of Palestinian and Israeli control. Overall, full sovereignty remains elusive, and many aspects of governance and security are influenced by external factors.
Firstly, Ehud Barak was a Prime Minister, not a President. He offered the Palestinians control over the entire Old City of Jerusalem (with a Jewish committee to oversee the Western Wall), something which has never been offered before or since to the Palestinians. This offer, although rejected by the Palestinians, created tremendous furor in Israel, leading to Ehud Barak losing the next election to Ariel Sharon in a landslide.
As of right now (May 2014), nobody is. Until a few weeks ago, Hamas and the Palestinian Authority were competing for political power in those areas, but they signed an agreement a few weeks ago, and they are presently making nice in public. In an overarching and general sense, the Israelis and the Palestinians are fighting over the Gaza Strip and the West Bank. However, the Israeli government has already ceded the claim on the Gaza Strip to the Palestinians and claims that it is willing to devolve control of most of the West Bank to Palestinians as well.
The biggest concern for most Palestinians is their future. They are worried that more and more illegal settlers will take over the land designated for the Palestinians. Settlers erect their homes legally according to Israeli views which would then drive the Palestinians away from their land.
Palestinian editorials rarely have anything nice to say about the Israeli military, often portraying the Israeli military as cruel and sadistic bastards who like nothing more than yelling and cursing at Palestinians, regardless of any Israeli soldier's personal feelings about his job. Israeli editorials are much more mixed. Right-wing editorials are often written for settlers who come into conflict with soldiers over their perceived right to settle wherever they wish and dispense "justice" like cowboys. As the hand of the law, the Israeli Army often has to curtail the activities of these rowdier citizens and this gets their ire. Therefore, they portray the Israeli Army as fifth column against the expansion and settlement of the Jewish people in accordance with prophecy. The left-wing editorials often see the Israeli military as a necessary evil, and often complain about what the Israeli system does to its young people. It often also mentions the difficulties many soldiers face when on patrol in terms of their emotional qualms with the Occupation (if they have any) and the instigation of both Jews and Palestinians in their purview.
If the intended phrasing was Israelis and Palestinians (i.e. both groups of people), there are an incredible number of disagreements between the Israelis and Palestinians and I invite you to read more about them here. If I had to choose the most important of those disagreements, it would be that both Israelis and Palestinians wish to each create a sovereign nation where they have power to create a dominant religious/secular, cultural, and legal systems and are unwilling to cede to the other specific key areas of the territory that would be requisite for that.However, the question specifically phrases it as Israel and the Palestinians, (i.e. Israel as country or the government of that country and Palestinians as a group of people). The Israeli government has no issues with Palestinians per se; there are over 1.6 million Palestinian citizens of Israel and around 300,000 Palestinian permanent residents. The Israeli government, however, has a vested interest in keeping Israel as a state with a Jewish majority in order to preserve its identity and so the desire that the worldwide 11 million Palestinian population has to live in the lands under Israeli sovereignty necessarily threatens that goal. This results in antagonism between the Palestinians and the Israeli government, which has come to blows at several points in the history of Israeli-Palestinian relations.
Control of land in the Middle East by Palestinians and Jews.
The West Bank (Jordan) and the Golan Heights (Syria), both taken in the Six Day War in 1967. The Gaza strip is no longer an occupied territory. The West Bank has three types of Areas: A;B;C. Area A - full control of the Palestinian Authority. Area B - Palestinian civil control and Israeli security control. Area C - full Israeli control, except over Palestinian civilians. While the vast majority of the Palestinian population lives in areas A and B, the vacant land available for construction in dozens of villages and towns across the West Bank is situated on the margins of the communities and defined as area C. The Palestinian Authority has full civil control in area A, area B is characterized by joint-administration between the PA and Israel, while area C is under full Israeli control. Israel maintains overall control over Israeli settlements, roads, water, airspace, "external" security and borders for the entire territory.
Jews and Palestinians.
The question and answers concerning Israeli policies and how they effect Palestinians is a long and complex situation that would and have filled many books. Currently, Palestinians living in Israel proper, have generous rights not afforded to peoples in other Middle Est nations. In the controversial West Bank, land that was acquired after the 1967 3 day war, live as people under the occupation of the IDF the Israeli army. There is much controversy concerning relations between Palestinians and Israelis that live in the occupied area called the West bank. There Palestinians are said by many to be treated fairly. Others have said that the Israeli occupation there is bad for all Palestinians.
Palestinians assert that they should retain control over East Jerusalem, which they consider the capital of a future Palestinian state. They emphasize the city's significant historical, cultural, and religious importance, particularly for Islam and Christianity. Additionally, Palestinians seek to ensure access to holy sites and maintain their demographic presence in the city. Any agreement to end the conflict would need to address these concerns to be acceptable to the Palestinian leadership and populace.
Generally, Yes. Specifically, No. The Arab-Israeli Conflict is primarily about which government (or governments) should be operating in the former British Mandate of Palestine. The overriding question is whether there should exclusively be a Jewish State, exclusively be an Arab State, be some sort of bi-national State, or if there should be two or more states where some are Jewish States and some are Arab States. The reason for the "Generally, Yes - Specifically, No" above is that many pro-Palestinians reject calling what is currently Israel-proper Israel and many Israelis, Palestinians, and others reject calling the West Bank and the Gaza Strip (which are also in contention) part of Israel.