Tl,dr; Zionists believe that all Jews have the right to live in a
Jewish state in historical Palestine (aka Land of
Israel). Anti-Zionists believe the opposite.
There are at least two brands of anti-Zionism; religious and
secular. Religious anti-Zionists believe that the establishment
of Israel displeased God and violated the Jewish people’s oath of
political quietism. Based on exegesis, they argue that Jews
should wait until the coming of messiah. An example of a religious
anti-Zionist group is Neturei Karta, depicted below:

Secular anti-Zionists believe that Zionism is Jewish settler
colonialism. Two prominent secular anti-Zionist groups are Jewish
Voice for Peace (JVP) and the BDS movement. Both have written
extensively on their views on anti-Zionism (see previous links), which
I'll summarize here.
Settler colonialism is the process during which one group of people
wrestle control over land from the hands of the indigenous
population. They remove or eliminate the indigenous people, erase
their culture, and replace it with their own economy and system of
governance. Examples of states created through settler colonialism are
New Zealand, Australia, Canada, South Africa, the United States, and
Israel. In each case indigenous people and culture was replaced with a
more powerful foreign people and associated culture. The main
difference between the colonization of Palestine and other regions,
they argue, is that the colonization of Palestine is ongoing as the
process began relatively late. In 1948 when Israel was declared,
European settlers had already supplanted the indigenous populations in
the Americas. BDS uses the following map which purports to show how
Zionist settlers have usurped more and more Palestinian land:

Similarly, former JVP Deputy Director Cecilie Surasky in her speech
Settler colonialism, white supremacy, and the “special relationship”
between the U.S. and Israel argues that Zionism is not unique, but
should be analyzed in a broader historical context:
I have seen these views manifest in the movement for Palestinian
liberation: sometimes people chant “2-4-6-8 Israel is a racist
state”, or decry the disappearance 400 Palestinian villages when
Israel was created, without even a hint of irony or self-reflection
that one is literally standing on land built on slavery and the
(still happening) genocide of indigenous peoples.
Her point is that anti-Zionism is about the indigenous Palestinians'
rights, but indigenous peoples' rights in other parts of the world are
equally important.
Back to your question:
But I had always understood zionism to mean the Jewish wish for a
homeland. This clearly isn't what the anti-zionists mean by the
word. What do they mean?
If you add "Jewish wish for a Jewish homeland in Palestine"
then that sure is what they mean. Omar Barghouti, co-founder of the
BDS movement, writes:
A Jewish state in Palestine in any shape or form cannot but
contravene the basic rights of the indigenous Palestinian population
and perpetuate a system of racial discrimination that ought to be
opposed categorically.
Just as we would oppose a "Muslim state" or a "Christian state" or
any kind of exclusionary state, definitely, most definitely, we
oppose a Jewish state in any part of Palestine. No Palestinian,
rational Palestinian, not a sellout Palestinian, will ever accept a
Jewish state in Palestine.
His alternative is a state with equal citizenship:
Accepting modern-day Jewish-Israelis as equal citizens and full
partners in building and developing a new shared society, free from
all colonial subjugation and discrimination, as called for in the
democratic state model, is the most magnanimous, rational offer any
oppressed indigenous population can present to its oppressors. So
don't ask for more.
This is an offer all Zionists reject. A democratic state with no
privileges for Jews, where Jews wouldn't even be a majority, would end
the Zionist project.