Israelis are citizens of the state of Israel.
Arab citizens of Israel are "Arab Israelis" or "Israeli Arabs" (the two phrases are interchangeable).
The term "Palestinian" can have several meanings.
Before 1948, Palestinian either meant Jew or specifically a Jew whose residence is in what is now Israel.
(for example -- The Palestinian Brigade was a Jewish army that fought in WWII)
In 1948, the Palestinians voted to become Israelis.
In 1964, the word Palestinian was taken by an anti-Israel Arab terrorist group to refer to those Arabs who live in what is currently Israel.
From 1964 onward it is thus a political term -- used by anti-Israel propagandists to refer to Arabs who are either residents of Israel or claim to have resided in the area of Israel prior to 1948.
When used by people to describe themselves, it is used by those Israeli Arabs who reject Israel, as well as by those Arabs living outside Israel who make a claim to land in Israel.
After the creation of the Palestinian Authority in 1996, the term Palestinian obtained the additional neutral meaning of "a citizen of the Palestinian Authority"
when the country of Gaza was created in 2005, the citizens of Gaza were likewise called "Palestinian" because theoretically Gaza and the Palestinian Authority are supposed to be one country with one government.