The definition of Zionism has evolved over time: There’s pre-1948 Zionism and post-1948 Zionism (not to be confused with post-Zionism, another idea altogether).
Post-1948 Zionism — arguably what most people consider the definition of Zionism today — can be simply defined as the belief that the State of Israel has a right to exist, that Jews have the right to self-determination. (This definition is up for debate, though.)
Pre-1948 Zionism is a little bit more complex. To sum, it was the general movement to establish a Jewish state. The modern state of Israel is therefore the culmination of Zionism, the Jewish effort to establish an autonomous state and end the diaspora of the Jewish people. Political Zionism was a product of many trends: the persecution of Jews in Europe and Arab lands; the rise of nationalism around the world; idealistic visions for building a new kind of society; and the conclusion that Jews would only be safe if they controlled their own destinies, to name a few.
Post-1948 Zionism — arguably what most people consider the definition of Zionism today — can be simply defined as the belief that the State of Israel has a right to exist, that Jews have the right to self-determination. (This definition is up for debate, though.)
Pre-1948 Zionism is a little bit more complex. To sum, it was the general movement to establish a Jewish state. The modern state of Israel is therefore the culmination of Zionism, the Jewish effort to establish an autonomous state and end the diaspora of the Jewish people. Political Zionism was a product of many trends: the persecution of Jews in Europe and Arab lands; the rise of nationalism around the world; idealistic visions for building a new kind of society; and the conclusion that Jews would only be safe if they controlled their own destinies, to name a few.