Theism: the belief in the existence of a deity. The theist is the person who makes the claim that a deity exists. Who or what the deity is is not important, just that the claim is being made.
Agnosticism: the belief that both the existence or nonexistence of a deity is either unknown or unknowable. An agnostic is a person who believes that it is impossible to know whether or not a deity exists. Contrary to popular belief, an agnostic is not someone who is between theism and atheism. It is also not a person who is undecided as to the existence or nonexistence of a deity.
Atheism: the rejection of the belief that a deity exists. As I'm sure you would expect, I'm going to go into this one into a little more length. An atheist is a person who lacks the belief that a deity exists. The atheist does not, however, turn around and claim that a deity does not exist. This is one of the most common misconceptions regarding atheism, and needs to be corrected. We atheists do not believe god doesn't exist, we just lack the belief that he does. I'll explain it with a couple of analogies in a second. Right now, I need to make this more confusing.
Some philosophers in the past decades have attempted to branch atheism into smaller parts. While there are others (implicit/explicit atheists for example), TheseThere exists two forms of atheists: positive and negative. Positive atheism is also known as hard or strong atheism, and is characterized as having the explicit affirmation that gods do not exist. A negative atheist, also called a weak or soft atheist, is characterized as having the belief that deities to not exist, but they are not exactly claiming that the statement is false. Many times, atheists will hold both of those forms of belief at the same time, but with reference to different gods.
Nontheism: a religious or nonreligious ideology that rejects belief in a deity. This is a broad group of beliefs or philosophies that reject belief in god.
Freethought: a philosophical viewpoint that holds that opinions should be formed on the basis of science, logic, and reason, and should not be influenced by authority, tradition, or dogma.
Skepticism: denote any questioning attitude, or some degree of doubt regarding claims that are elsewhere taken for granted. Essentially, this is a group of people who are skeptical of common beliefs.
Humanist: a rationalist who believes that humanity is capable of morality and self-fulfillment without reliance on supernaturalism.
Apatheist: a person who thinks the question of God’s existence is irrelevant and unimportant.
New Atheist: a person who not only is an atheist but believes that religion is, on the whole, harmful and should be opposed whenever it conflicts with science or threatens societal interests.
Secularist: a person who opposes the injection of religion into civil affairs, particularly public education. Can also mean a person who rejects all forms of worship and religious faith.
Rationialist: an individual who relies on logic and reason for knowledge and a system of ethics, rather than on faith or religion.
To explain a couple of these positions a little more clearly, a good analogy is a courtroom. The theist is like the plaintiff. They are the ones claiming that the person is guilty (analogous to "god exists"). The defendant however, is claiming innocence (similar to "god doesn't exist"), and is therefore the nontheist. The other categories are in the jury, because only the theist and nontheist are actively making claims about something. The atheist is the person in the jury who votes "not guilty". Note that the atheist is not voting "innocent", i. e. there is no god. He is simply saying that there isn't enough evidence to vote guilty (pro-god). The agnostic is the person in the jury who would not vote due to lack of evidence both for and against, and just leave the room in an angry fashion.
I hope that little analogy works. It helped me get a hang of the little differences.