Why did it begin?
When Britain occupied Palestine during WWI, they gave a declaration to the Jews that they will give them a homeland in Palestine. There was one big problem however: the land already was populated by indigenous Arabs (Mostly Muslim, but with significant Christian 6% and Jewish minorities 10%). This obviously angered the indigenous Arabs. They also wanted this land.
Why did it continue?
The Israeli-Palestinian conflict refuses to be resolved because it is a conflict unprecedented in human history. It is no longer an argument between land, but more so the people. There have been so many events in which ministers and presidents have been assassinated, due to the protests and unrest.
Key issues that must be addressed for it to end?
The remaining key issues are: mutual recognition, borders, security, water rights, control of Jerusalem, Israeli settlement, Palestinian freedom of movement, and finding a resolution to the refugee question. The violence resulting from the conflict has prompted international actions, as well as other security and human rights concerns, both within and between both sides, and internationally.