Why can't Israel live in peace?

What is the source of the conflict between Israel and the Palestinians?

There is no casual answer to this question. For the long answer, I recommend Sister Ruth Lautt's comprehensive article, The Church’s Witness on Issues in the Arab/Israeli Conflict. My shorter answer is below.

The land we call "Palestine" has never an independent nation since before the time of Christ, having since then been ruled by the Greeks, the Romans, the Byzantines and Arabians, briefly by Europeans during the Crusades (and not all of Palestine then). Until the establishment of modern Israel in 1948, the people living in historic Palestine had never been self governing since the land belonged to ancient Israel/Judah.

In late modern times, most of the Middle East was under the sway of the Ottoman Empire, headquartered in Turkey. The Ottomans made the unfortunate decision to ally with Germany during World War I. The end of the war meant the end of the Ottoman Empire, and Palestine came under control of the British as a prize of war. (British forces under Edmund Allenby took Jerusalem in 1917.)

The land that became modern Israel was sparsely populated at that time. One-third of the people living there were Jews, more than had lived there since Roman times. The League of Nations, formed in 1919, passed resolutions directing the British to create a Jewish homeland. This directive became known as the British Mandate and ultimately came to include approximately the lands that became Israel in 1948, plus the Transjordan (known today as the West Bank) and Gaza.

In 1947, the United Nations resolved that the lands of the British Mandate, not including Jordan, be partitioned into two states, one Jewish, one Arab. The Jews accepted this resolution while the Arab nations violently opposed it. The result was the war for Israeli independence, beginning in May 1948 and culminating in a ceasefire in February 1949. During the war hundreds of thousands of people, both Jews and Arabs, were dispossessed of their homes and forced to leave or chose to leave because of the violence of war. Refugees were Arabs and Jews in approximately equal numbers. Sister Ruth summarizes,

The Arabs that stayed in what became the borders of Israel became Israeli citizens. The Arabs that fled or were forced out became refugees. For the most part they were never resettled and the United Nations maintained and continues to maintain them as refugees in Lebanon, Syria, Jordan and in what we now call the Palestinian territories under a special agency created only for Palestinian refugees -- United Nations Relief and Works Agency for Palestine Refugees in the Near East (UNWRA).

Today, these refugees and their descendants live mainly in the West Bank and Gaza. Neighboring Arab countries have been very inhospitable in accepting them. For example, after the Gulf War of 1991, Kuwait expelled all Palestinians living and working in the country, about 200,000 overall.

Since 1948, the Arab nations have not accepted the legitimacy of Israel as a state and have rejected the concept of a "two-state solution," the basis of the original British Mandate that would have created both a Jewish nation and a Palestinian nation. The two-state solution still forms the basis of the West's attempts to mediate a peace. This is exactly the solution that Hamas rejects. Even Yasir Arafat, when proclaiming his acceptance of the state of Israel to the West, made it clear in his domestic statements that Israel could not continue as a Jewish state, but would have to become a majority-Muslim country.

Except Egypt, no Arab nation has formally signalled acceptance of Israel as a Jewish state, although Jordan does so in practice if not in declaration. Most of the rest of the Arab countries deal with Jewish Israel as a fait accompli, though they look forward to the day when it will transition to Muslim majority.

This history is the root of the conflict between Hamas and Israel (and, for that matter, between Israel and Hezbollah in Lebanon).

Update: See also my later post, What does Hamas aspire to?