Put it plainly, as Zionism is the belief that Jews have a right for self-determination anti-zionism is the belief that Jews, unlike every other nation, does not have that right.
That does not sound progressive and cool so people who declare themselves anti-zionists have a variety of explanation, or rather excuses.
Some of the most popular ones I've encountered.
Historical anti-Zionists
Some say that back at the end of the 19th century and beginning of the 20th century there were many Jews (e.g. the Bund or "General Jewish Labour Bund") party that was against Zionism. These people tend to pretend that not wanting to create a Jewish state 150 years ago and wanting to abolish the existing one today are two totally different things. So people who say it is ok to be anti-Zionist now as some Jews were anti-zionists 150 years ago seem to disregard that time changes things.
Judaism is only a religion anti-Zionists
Some say that being Jewish is a strictly a religion and religions don't have the same "right" to have their own country. These people seem to want to deny what the Jews have been saying and thinking about themselves for more than 2000 years. That they are a nation. These people also seem to totally disregard the millions of Jews who are not religious at all. Earlier I looked at countries and religions and specifically found it extremely interesting that on Wikipedia on the page about Israel in the "religion" section they have "Judaism" and under Judaism they have "Hiloni". In other words Jewish atheists are stil categorized as religious Jews. Whatever...
These people also tend to disregard the fact that there are tens of countries that are defined as Muslim or Christian. They never seem to raise an objection against the existence of the countries.
Redefine Zionism anti-Zionists
Some people plainly redefine Zionism as something horrible and then they say they are against that horrible thing so they are anti-zionists. I've heard these people define Zionism as the movement (by Jews) to subjugate every else. As the movement to kill Arabs. Some of these people say that the goal of Zionism is world domination etc. etc. You get the idea. I would call them lunatic, but some of these people are quite normative in their other conducts in the world. They have just found (or created) some definition under which they believe they are doing the "right thing".
This is a strange bunch of people. They usually say they are not against the existence of Israel they are only against those other things they think (or pretend) zionism means. However, I am not sure if they only say this as this sound more PC or if they really believe in it.
Neturei Karta anti-Zionists
Neturei Karta is a very small (1-2000 people) extremist Jewish sect that supports the destruction of the state of Israel.
The funny thing about the people who parade with Neturei Karta and try to use them as a prove that even Jews don't think that Israel should exist is that Neturei Karta believes that the are will be Jewish once the Messiah comes. So effectively they are only against the timing of the creation of the Jewish state.
Conclusion
It is funny and sad to see how much energy and intellect some people invest in trying to be against Israel.