The modern conflict I believe you're referring to essentially began in 1948, when Israel was formed as a nation by the UN and it formally declared independence. It was immediately attacked by five nations, and in the ensuing war Israel won, and expanded its border to include most of what had been called Palestine.
Some Palestinians chose to leave and others were forced to leave, and most of them became refugees. With no homeland and feeling they had been cheated out of theirs, the struggle began as Palestinians used both violence and public pressure to try and reclaim the land of Palestine. Israel responded sometimes with force and other times with US help in the United Nations to defend its independence as Israel.
So the modern conflict has been going on now for almost 62 years.