To understand the roots of the Arab-Israeli conflict, you need a bit of historical context as to the history of the region. After being conquered by the Romans, there were a number of Jewish revolts against foreign rule (the last of which ended in 136 CE), which ultimately culminated in the vast majority of the Jews either fleeing the region or being sold into slavery. The ones that were left were now a minority in their own homeland.

Fast forward 1,800 years to the 19th century, and we have a large movement among the Jewish diaspora who wish to create a "Jewish homeland" where the state of Israel was (which has been part of various Muslim empires ever since the Byzantines were forced out in the 7th century CE). Following various pogroms, purges, and the rise of Herr Hitler himself, there were large Jewish migrations from various parts of the world to Palestine. By the end of WWII, they comprised 33% of the population.

In the aftermath of WWI, the Ottoman empire had collapsed and the British and French had effectively divided everything between Turkey, Iran (then the Persian Empire), Egypt, and the Caucasus between themselves. The area that we now know as Israel was the British mandate of Palestine.

To cut a long story short, after WWII various Arab and Jewish groups started covertly warring with each other, all with the British trying (and failing) to keep the peace, and attempting to prevent mass Jewish immigration from Europe.

A war-weary Britain decided that they'd had enough of Empire in general, and had been in the process of offloading their colonial possessions since the end of WWII. After a few failed attempts at partition Palestine through peaceful means, the British evacuated Palestine and terminated their mandate on the 14th of May 1948. That afternoon, the state of Israel declared its independence. The next morning, Egyptian, Lebanese, Syrian, Jordanian, and Iraqi troops invaded.

When the war finally ended in March 1949, not only had Israel survived, but they'd actually gained territory, to the surprise of both the British Foreign Ministry and the CIA. Ever since, there have been numerous attempts by the Arabs to force the Israelis back into the sea (the 6 day war in 1967, and the Yom Kippur war in 1973), and numerous organisations have been created with the goal of restoring the Palestinians to their homeland (the stated aim of both the Palestinian Liberation Organisation and Hezbollah is the elimination of the state of Israel).

So in summary, the Arab-Israeli conflict revolves around the state of Israel's right to exist. The Arabs claim that it's the homeland of the thousands of Palestinians who were forced to leave in the aftermath of the war of independence, whereas the Israelis can claim the same thing. It hasn't been resolved (and is unlikely to be resolved in the near future) because the only outcome that would truly end the conflict would be the complete genocide of the world's Palestinians or Jews. Fundamentalists on both sides are completely unwilling to accept any outcome other than total victory. The aims of both groups are almost irreconcilable.

U/tayaravaknin is quite knowledgeable on the topic. His thread here has a lot of information that can go a long way to answering your question, and I know he is fairly active if you have any specific questions. I should point out that getting too far into the present would probably break the sub's minimum limit to discuss history (the 20 year rule).