(These are not meant as formal or scholarly definitions, but only definitions of the terms as I understand them).
Theist: Someone who believes in a deity or deities. This may be qualified with a particular name, such as "Christian theist." This term seems to be used a lot less often than the corresponding term "atheist." I've seen it used mostly as a catch-all term for believers, when the person in question wants his listeners to be sure that he doesn't mean, say, believers in a philosophical system.
Agnostic: Somewhat a confusing term, as technically it refers to a state of knowledge rather than a state of belief. (Theism and atheism tend to refer to belief). Someone who doesn't know if a god or gods do(es) or do(es) not exist. In the broadest sense, all humans are agnostic, since very few people claim to "know" that god does or does not exist. The two terms that follow are also used, and may be more useful.
Theist agnostic: Someone who admits to a lack of certain knowledge as to a deity's existence, but still believes, proof or no proof. Many theists may actually be theist agnostics, but few would state it in those terms.
Atheist agnostic: Someone who admits to a lack of certain knowledge as to a deity's existence, but does not believe (often because of a lack of evidence). This term tends to be used more often than "theist agnostic," and often by those who wish to make a declaration as to knowledge as well as faith. Will sometimes overlap with "weak atheist," below.
Atheist: Someone who lacks a belief in god(s). This can take place along a spectrum (two varieties are mentioned below), but rarely, if ever, does it take the form of a denial of gods. After all, why would an atheist bother to deny something he doesn't think exists? It would be like denying the Tooth Fairy.
Weak atheist: Someone who doesn't believe that gods exist, often because of a lack of positive evidence as well as faith. This position has no burden of proof, since the statement "I don't believe that gods exist" does not make a positive assertion. Theists confronting this type of atheist should be willing to offer proof. I call myself a weak atheist.
Strong atheist: Someone who believes that no gods exist, and thus makes a positive statement. Strong atheists usually believe they can offer proof of the non-existence of gods, perhaps from science, perhaps from observation of the natural world, perhaps from contradictions in a particular religion's belief system. They may have come to their atheism any number of ways, just as a weak atheist might, but may be stronger arguers.
Implicit atheist: This term is discussed in George Smith's book Atheism: The Case Against God, an excellent beginning source, though now somewhat dated, to ground certain atheist concepts. While a large part of the book is specifically an examination of the arguments that the Christian religion has used historically, there's an excellent introduction that explains the differences between various kinds of non-belief, and uses the term "implicit atheism." In Smith's view, an implicit atheist is one who has no god-belief because he or she has never been exposed to the concept. This might include someone reared in an environment completely apart from religion; it also includes children before they hear of the god-belief. I find this argument compelling, because I think it says something that faith in a particular god gets trained in through the culture, rather than simply appearing in the mind as it might if the god were natural and external to the human mind.
Explicit atheist: Smith also discusses this term. An explicit atheist is one who has heard of and rejected theism, perhaps through deconversion, perhaps casually as a child. Under these terms, I am an explicit atheist, as is almost everyone, since it would be extremely hard to grow up in a theistic culture and not hear about it.
Anti-theist: This is the term that many people mistake atheism for. An anti-theist is one who hates and attacks theism and theists. This label by no means fits all atheists. There may be bitter atheists who attack theists. On the other hand, there are bitter theists who attack atheists, and no one labels them "anti-atheist."
