anarchists are against the state, which means that we are against a kind of (frozen) power that today is best exemplified by governments. we are also against capitalism, which is a system of values and practices that quantify everything.
people who call themselves anarcho-communists are focusing on the communism that marx posited as the end of history - the "goal", the post-revolutionary time, and the "anarcho" refers to how they think people will get there.
there is confusion over communism as posited by marx, and communism as it has been claimed by state powers (russia, china, etc). some people do not call themselves communists mostly because the history is so horrible for the word, that it is more work to reclaim it than it is worth.
people who call themselves communists want communism but don't usually believe that people will get there without coming together in big bodies, and may or may not believe that communism will be achieved through a state.
because anarchists are so widely variant, there are plenty of anarchists who also believe that people need to come together in big groups in order to create change. so there is a lot of overlap in practice as well as ideas between some anarchists and some communists. this is made more complicated by the historical tendency of communists to act like the friends of anarchists in times of conflict, and then to kill us (see the spanish civil war and kronstadt, among others).
socialism has also been sullied by the real world experience of people who call themselves socialist, but to the extent that it means "collective ownership" (*not* state ownership), some anarchists have used it. other anarchists reject the concept of ownership (as currently understood) altogether, as part of the training we have gotten from capitalism.