Do you mean the war in 1948 or in 1967?

The Arab-Israeli war of 1948 is a complex conflict, and was preceded by significant amounts of fighting in the form of a civil war, riots, and more. To explain all of this would take a book. I can help you understand a bit, but you'll have to be more specific if you have follow-up questions!

The Arab-Israeli War formally began in 1948. But it had been, as I said, brewing for quite some time. Starting with the Zionist movement in the 1880s (though some thought and time was devoted to this beforehand) there developed a movement among Jews to return to their homeland of Israel, held at the time by the Ottoman Empire, and to have a state of their own. This movement began with groups like Hovevei Zion (Lovers of Zion), who organized what were known as "Aliyot" where Jews from places like Eastern Europe or Russia (among others) would move to what is now Israel, purchase land, and begin working the land and laying the groundwork for a state of their own. This movement was, at first, extremely unsuccessful. The people who moved there lived hard lives, and most left within the decade after arrival. One traveler wrote in 1885:

Nobody knows of all the hardships, sickness, and wretchedness they underwent. No observer from afar can feel what it is like to be without a drop of water for days, to lie for months in cramped tents visited by all sorts of reptiles, or understand what our wives, children, and mothers go through when the Arabs attack us...No one looking at a completed building realizes the sacrifice put into it.

While this "First Aliyah" was largely unsuccessful, further Aliyot followed. The Second Aliyah, a wave from 1904 to 1914, followed the beginning of Zionism in earnest. This came after the founding father of political Zionism, a movement that sought to use diplomatic means to convince superpowers to grant Jews permission to establish a state as well as continue the settlement process, published his famous pamphlet in favor of a Jewish state, Der Judenstaat (The Jewish State). The founding father, Theodor Herzl, convened the first World Zionist Congress, which was a gathering of Jewish intellectuals and leaders who agreed to work together towards establishing a Jewish state.

This process continued, with more Jews arriving in Israel. The Ottomans refused the Jewish request for a state, but lost control of the area during WWI to the British after their empire fell apart. The British assumed control and got a stamp of legitimacy from the League of Nations (more of a rubber stamp, to be frank), which endowed them with the "Mandate for Palestine". The British did not grant Jews a state either, though they promised one, and some argue they promised the Arabs of the area one as well, in a conflicting manner (read the Balfour Declaration and McMahon Correspondences). At any rate, as more Jews arrived and sought to set up a state, tensions increased, and violence broke out, with Arab riots in 1920, 1921, 1929, and a full-blown revolt against the British in 1936, which lasted for around three years.

Following the end of WWII, Jewish groups were particularly strong, while outnumbered by Arabs in the area. There were three groups mainly: Haganah (by far the strongest), Irgun (weaker than Haganah, but middling in power), and Lehi (weakest). However, the smaller the group, the more radical it was. Lehi, for example, briefly considered seeking an alliance with the Nazis during WWII, and assassinated British politician Lord Moyne in 1944, and a Swedish diplomat who was trying to negotiate a peace deal in the Arab-Israeli war in 1948. Irgun was willing to hit more civilian targets and accept more collateral damage, while Haganah was more focused on disrupting command and control and supply to British troops. All three sought a Jewish state, but in different ways.

At any rate, this setup, as well as the previous Arab revolt and the effects of WWII, all contributed to the British decision to withdraw from the Mandate. This decision meant that the successor to the League of Nations, the United Nations, effectively had the problem punted over to them. You see, for a decade the British had been trying to come up with plans for one state or two states (one Jewish and one Arab), and no plan had succeeded in being accepted by both sides (sometimes, it was rejected by both sides, sometimes only one would accept a plan in principle but not in detail, etc.). So the United Nations created a special task force of neutral (or as close as they could get to neutral) countries, called the UN Special Committee On Palestine (called UNSCOP for short), which was to travel to the area and study the problem and propose solutions.

UNSCOP came back with two plans: one endorsed by 8 member countries, and one endorsed by 3. The majority plan is what we know as the UN Partition Plan, which was endorsed and recommended by the UN General Assembly on November 29, 1947. That plan proposed a Jewish state and an Arab state. Neither side was fully happy with this proposal (I can explain why if you'd like), but the Jewish side reluctantly accepted it, and the Arab side did not. The minority report, which proposed a federal unitary state, was not endorsed by the UN.

Following the passage of the partition plan, however, violence broke out. Arab and Jewish paramilitary forces began a civil war, with Arab forces getting aid from "volunteers" and in the form of supplies from the surrounding Arab states. However, their disorganization and the lingering effects of the revolt's failure in 1936 had left them weaker than the well-organized Jewish forces, which had all but won the civil war by May 1948. The British were withdrawing, and had previously announced that the British Mandate would formally expire on May 15, 1948, and their troops would be fully evacuated by August 1948. So on May 14, 1948, the Jewish leadership had all but won the war, and saw Arab states getting ready to intervene the moment the British Mandate formally expired. So they declared independence, proclaiming the creation of the state of Israel, effective at midnight on May 15, 1948, the moment the Mandate expired.

The new state was established, but the Arab states had to invade. The Jewish goals were relatively simple: create a state, hold onto it, and seek to control as much land as possible from the former British Mandate, particularly areas with more Jews. That was the main goal of the Jewish leadership. Subsidiary goals are argued over all the time. For example, during the war, expulsions and fleeing led to roughly 710,000 Palestinian Arabs losing their homes and going into neighboring Arab states, and into the Jordan-occupied West Bank and Egyptian-occupied Gaza (the two states took control of those areas and did not let them go, but lost them to Israel in the 1967 war). There are those who question how high up the desire for these expulsions and fleeing went, and whether or not they were a main goal to tip the demographic balance towards Jews. However, it appears that this goal was not a central one made at the high levels of Jewish leadership, and that it was instead made by individual commanders in the field: the commanders in the South expelled far more people than those in the north, for example. At any rate, the secondary Jewish goal that we can know was to create a safe haven for Jews from worldwide to have, to escape from anti-Semitism. Hundreds of thousands of Jews, for example, were sitting in displaced persons camps following WWII and the end of the Holocaust, and Israel sought to take in as many as it could, if it could keep its state alive. Israel also knew by the time of the war that Jews in Arab states would likely not be safe if it won the war, and prepared to accept those Jewish refugees as well (roughly 850,000 Jews would flee or be expelled from Arab states in the decade following the war). Israel's Jewish population virtually doubled in its first three years because of this influx of new Jewish arrivals from Europe and the Arab world.

Does that more or less answer your question? Jews wanted a state of their own, did not launch the Arab-Israeli war, and sought to defend the state they created.