An atheist lacks belief in a god or gods. That's it. Lack of belief.
Anyone who has never heard of the idea of a god is automatically an atheist. The term implicit atheist is used for this, though, to distinguish them from people who have at least thought about the idea of a god first. All children are implicit atheists.
People who adhere to non theistic religions are also technically atheist, although the term nontheist is more commonly used for this. This can include religions like Buddhism which don't have a deity, or world views like Deism and Pantheism.
Deists believe in an intelligent supernatural creator of the universe, but believe that this deity does not observe or intervene in the world. They do not believe in miracles or revelation, but instead hold that the Creator reveals itself in the natural world, and can possibly be known through human reason. _Pantheists _believe that god is essentially a synonym for the universe and reveals itself through nature.
Theists believe in a theistic deity, a personal god (or gods). A theistic deity is commonly held to be an intelligent supernatural being that created the world/universe and which has a personality. A theistic deity directly observes and intervenes in the world. This is usually manifested as watching over people, listening to their prayers, providing revelations, performing miracles and having a "personal relationship" with people.
Atheist is the lack of belief in a theistic god or gods.
Atheists don't (necessarily) believe that no theistic gods exist anywhere. A minority of atheists believe that and it is known as positive atheism or strong atheism. The term weak atheism or negative atheism is often used as a contrast.
An atheist says "I do not believe there is a god". A strong atheist says "I believe there is no god".
Some people use the term agnostic to mean atheist and use atheist to mean only positive atheist, which makes things a bit confusing. I don't like this scheme, since theism/atheism is an ontological position and gnosticism/agnosticism is an epistemological postion. Ontology deals with the true underlying nature of reality, wheras epistemology deals with our means to know it.
An agnostic says that we cannot know for sure whether or not there is a god. It's to do with whether we have knowledge/proof and how and whether we are able to obtain such knowledge, not the underlying reality of whether there is a god or not. To complicate matters further, there are actually different types of agnostic. There are those who believe we can know but we don't know right now (weak agnostic), and others who believe that in principle we cannot ever know for sure about a deity (strong agnostic).
Many (possibly most) atheists are agnostic. Many theists are also agnostic, in that they are not certain about the existence of their deity, but they believe it exists nonetheless (i.e. they have faith in their deity).
I've tried to put the two dimensions together in the table below:
|Ontological claim||Theist||I believe there is a god and I can know that for sure.||I believe there is a god but I can't know for sure.|
|Atheist||I don't believe there is a god and I can know for sure either way.||I don't believe there is a god, but I can't know for sure either way.|
|Strong Atheist||I believe there is no god and I can know that for sure.||I believe there is no god, but I can't know for sure.|
I'm the first to admit that this is a simplification. I have used agnostic in the strong agnostic sense of not being able to know the existence of a deity in principle. I have also phrased the description with god in the singular (presumably in reference to any monotheistic deity) but it applies equally to polytheistic beliefs.
Atheism is not a dogma, nor is it a religion. Atheists just think that it is not sensible to believe in things for which no good evidence exists. An atheist could become either a strong atheist or a theist if there was evidence to warrant either belief.