In terms of categories perhaps this analogy will help: Out of all objects in the universe only a certain proportion of them are symmetrical, these objects need to have a very specific criterium to qualify as such. All things that are not symmetrical are asymmetrical. The same applies to all words with the prefix "a". Theists are people with a specific thing and atheists are everyone else, everyone who lacks that specific thing. So it includes people with no beliefs in god and people who believe that there is no god and people who believe
Now the agnostic issue that someone brought up before: It can be a difficult concept for some people to grasp especially when steeped in the platonic tradition of epistemology as most Christians are (ie. knowledge constitutes beliefs that happen to be true). The gnostic/agnostic position is a response to a specific epistemological proposition: it's quite important to not confuse this with belief. Do you have knowledge of God or do you not? That's the question, yes or no. Agnosticism is absolutely not a position in between theist and atheist, it is about knowledge, not belief. I think Diocle gave a perfect illustration on how these words interact. Belief and knowledge are separate categories, working independently but in parallel. So we can have agnostic-theists who admit ignorance but still believe, we have gnostic-atheists who think they know that there is no God, gnostic-theists who know that God exists (like our good friend Basics here) and agnostic-atheists who lack both knowledge and belief.
It's this last category of atheists who make up the vast majority of atheists, but within this sub-group there are further ways to break them down. There are those who believe that there is no God (but they do not comprise all atheists) and there are those who simply do not believe and additionally there are those who believe that theism is dangerous, these are antitheists (these can also be gnostic-atheists obviously). These groups can be broken down even further, looking at Eastern forms and indigenous forms, but I have the sense that I've already become boring, so I'll stop now.
I'll just say one more thing: I'm not saying that there aren't dogmatic atheists, there are. I'm saying there is no atheist doctrine, except for the one invented by Christians, which is basically just a lazy insult.