The United Nations created the State of Israel after WW2, in 1948, as the Jewish national home. Approximately half of the territory of Palestine was given to Israel and the other half to Muslim Palestinians.
Muslim countries rejected this decision, and immediately, Egypt, Jordan, Syria and Iraq invaded the territory, starting a war. Israel won the war, and the Palestinian territory was reduced to half its original extension and divided in two territories: the Strip of Gaza and the West Bank.
A Palestinian nationalistic group called Al-Fatah, began to launch different types of terrorist attacks in different parts of the world.
In 1967 there was another war between Israel and an Arab coalition. Israel won and took Gaza and Sinai from Egypt (Sinai would be given back a few years later), the West Bank from Jordan (including Eastern Jerusalem) and the Golan Heights from Syria. I think the UN considers these as occupied territories, but Israel claims that they are vital for their own security.
Then, in the 1990s a peace treatment was signed with Al-Fatah. They renounced to the use of terrorism and acknowledged the existence of the State of Israel; in exchange, they could create an independent Palestinian Authority in the West Bank and Gaza, with a sort of democratic regime.
In 2005, Israel withdrew all of its settlers and soldiers and dismantled its military facilities in the Gaza Strip, but continues to control its access. In 2006, the Islamic Resistance Movement, HAMAS (considered a terrorist group by some countries), won the election and took control of the Strip of Gaza, while the Palestinian Authority controls the West Bank.
Israel is supported by the King of the South. The Palestinians are not openly supported by any countries nowadays, but they are somehow connected to the King of the North.
This is very complicated and I might be wrong in some details, but in broad terms, this is the background of the conflict.