1. Israel has the better part of the territory in the area, and they won it from numerous aggressors among the Arab nations. They are not supposed to settle it under international law, but that's one of the major reasons they believe they have a right to the land. There is also the religion itself which defines modern day areas considered "Palestinian" as part of Israel according to the bible.
Palestine's legitimacy in the area comes from the fact that they have been living there for a long time before the Jews returned to the holy land. They don't want to be a part of Israel and believe they have a right to an independent state, as well as to defend themselves from Israeli encroachment on what they consider to be their land (often disputed territory).
2. Israel has launched numerous military assaults in the area, mostly to root out extremists, but sadly, many innocents are killed as collateral damage. They have also repeatedly refused to stop settlement of East Jerusalem, despite mutual acknowledgement that it was a critical part of restarting the peace process.
Palestine has proven a haven for terrorists who often kill or kidnap Israeli targets, prop up a government which supports violent action against Israel, and unfortunately, cause a lot of harm to Israel, even within the legally defined scope of its territory.
3. Hamas has greatly aggravated the situation, because as was mentioned before, they support violent action against Israel, and their military arm often launches missiles and other projectiles into Israel from the Gaza strip.
A couple extra notes:
Israel WAS given much of the land they have by Britain in 1948 due to the horrors of the holocaust. They are legally there. However, they were almost immediately attacked by their Arab neighbours, which they held their own against, actually capturing territory. This is where the issue of Palestine comes in, as the majority of it is contained in these captured areas. There are of course some issues with Israel's actual borders, but the major source of problems comes from what has been going on in captured territories like the Gaza strip.
I've tried to remain neutral. Careful what you hear, this is a VERY contested issue.
*bible ---> Torah, whatever the Jewish holy book is called.