What is an Atheist?
An atheist is anyone who doesn't happen to believe in any gods, no matter what their reasons or how they approach the question of whether any gods exist. This is a very simple concept, but it's also widely misunderstood. For that reason, there are a variety of ways to state this. Atheism is: the lack of belief in gods, the absence of belief in gods, disbelief in gods, not believing in gods.
The most precise definition may be: an atheist is anyone who does not affirm the proposition "at least one god exists." Although it may seem convoluted, it has a number of important elements: there is a proposition, it's not a proposition made by atheists, and being an atheist requires nothing active or even conscious on the part of the atheist — all that's required is not "affirming" a proposition made by others.
What is an Agnostic?
An agnostic is anyone who doesn't claim to know for that any gods exist or not, no matter what their reasons or how they approach the question of whether any gods exist. This is also a simple concept, but it may be as widely misunderstood as atheism is. One major problem is that atheism and agnosticism both deal questions about the existence of gods, but whereas atheism involves what a person does or does not believe, agnosticism involves what a person does or does not know. Belief and knowledge are related but nevertheless separate issues.
There's a simple test to tell if one is an agnostic or not. Do you think you know for sure if any gods exist? If so, then you're not an agnostic. Do you think you know for sure that gods do not or even cannot exist? If so, then you're not an agnostic. Everyone who can't answer "yes" to one of those questions is a person who may or may not believe in one or more gods, but since they don't also claim to know for sure they are agnostic — an agnostic theist or an agnostic atheist.
More: Agnosticism Basics for Beginners
What is an Agnostic Atheist?
An agnostic atheist has two qualities: they don't happen to believe in any gods and they don't claim to know or sure that no gods can or do exist. Not believing that some claim is true while also not claiming to know for sure that it's false is not only easy, it happens in lots and lots of different topics. It would be a surprise if it didn't happen when the topic is the existence of gods.
For some strange reason, though, many people have he mistaken impression that agnosticism and atheism are mutually exclusive. But why? There's nothing about "I don't know" which excludes "I don't believe." On the contrary, not only are they compatible but they frequently appear together because not knowing is frequently a reason for not believing. It's often a very good idea to not accept some proposition is true unless you have enough evidence that would qualify as knowledge.
What is an Agnostic Theist?
An agnostic theist has two qualities: they believe in the existence of at least one god and they don't claim to know for sure that this god or gods definitely exist. Believing that some claim is true while also not claiming to know for sure that it's true is not only easy, it happens in lots and lots of different topics. Why shouldn’t it also happen when the topic is the existence of gods?
Once again, many people have the mistaken impression that agnosticism and theism are mutually exclusive. They are convinced that agnosticism is some sort of "middle way" or "third way" between atheism and theism — but why? There's nothing about "I don't know" which logically excludes "I believe." Sometimes it perhaps should because there are some claims that you shouldn't believe without sufficient evidence to qualify as knowledge. Being a juror in a murder trial would be a good example.
Atheist vs. Agnostic: What's the Difference?
By now, the difference between being an atheist and being an agnostic should be pretty clear and pretty easy to remember. Atheism is about belief, or specifically what you don't believe. Agnosticism is about knowledge, or specifically about what you don't know. An atheist doesn't believe in any gods. An agnostic doesn't know if any gods exist or not. These can be the exact same person, but need not be.
Every agnostic is also either an atheist or a theist because any given belief is either present or not — there is no alternative to those two options. An agnostic may be unsure whether atheism or theism is more reasonable. An agnostic may not consider their theism or atheism very important. But regardless, belief that the proposition "at least one god is true" is present or not in everyone.