There are lots of ways to define these terms, to the extent that you can't be sure how people are using them unless you ask.
The general population of English speakers understands atheism to be equivalent to what Michael Martin calls "positive atheism" and what used to be commonly known among Internet atheists as "strong atheism"--an active disbelief in the existence of gods. That's a position which does have a burden of proof over mere nonbelief, also known as weak atheism or negative atheism. George H. Smith made the same distinction using the terms explicit vs. implicit atheism. Richard Dawkins complicated matters by redefining "strong atheism" as absolute certainty that there is no God (position 7 on his scale). I wish he had chosen a different term, as I think it's a mistake to associate positive atheism/strong atheism with certainty, proof, or even knowledge.
I used to like this distinction, but am less enamored with it because "weak atheism" or "negative atheism" or atheism as mere lack of belief in gods has a few logical problems as a basis for anything. A lack of belief is not a position, it cannot be used to motivate action or to infer conclusions from. Those who say that they are only atheists in the weak sense, however, do join groups and appear to draw inferences and conclusions as though they are using the nonexistence of gods as a premise, which means either that they are really implicitly using strong atheism as a position, or they are drawing those inferences based on other meta-beliefs.
The advantage of equating atheism with weak atheism is that theism and atheism then become contradictories which cover the entire space of logical possibilities--you either have a belief in one or more gods, or you don't. Under that definition, there's no space for agnosticism except as a subset of one or both of atheism and theism.
The definition of "agnosticism" that was given earlier in this thread as pertaining to the possibility of knowledge about the existence or nonexistence of gods then gives you two dimensions, on which you can have agnostic atheists (I don't believe there are gods, and it's not possible to know), agnostic theists (I believe in at least one god, but it's not possible to know), gnostic atheists (I don't believe there are gods, and it is possible to know there aren't), and gnostic theists (I believe there's at least one god, and it's possible to know). Of those positions, I think agnostic theism is difficult to make a case for with respect to most conceptions of God, except for deism and other forms of uninvolved gods.
But most people who call themselves agnostics aren't using that definition, they're using a notion that is a particular form of weak atheism, that holds to something like there is a parity between arguments for and against the existence of gods, or that there is no way to effectively compare their evidential weight, or similar. They might agree with agnosticism regarding the possibility of knowledge for the existence or nonexistence of gods, but they go further and say that there is some parity on the case for mere belief in either direction, as well.
I'm generally in favor of allowing people to choose their own self-identifying terms and defining them as they see fit, so long as they can give a legitimate reason for their classification and it's not completely at odds with ordinary usage. One example that goes beyond ordinary usage and I think just indicates some kind of confusion is that 21% of self-identified atheists in a Pew survey reported last October said that they believe in God. Sorry, but that's not a definition of atheist that I think can get off the ground.
My own position is strong atheism/positive atheism with respect to most traditional conceptions of God, and weak atheism/agnosticism (or igtheism) with respect to certain rarefied/unempirical notions of God. I'm comfortable calling myself an atheist in general, and dispute claims that it's impossible have knowledge that at least most gods do not exist. "You can't prove a negative" is a widely expressed canard, which I argue against here:
That also contains links to a few other essays which make the same point in a way that is probably more clear, including one by Jeff Lowder which argues for the possibility of disproofs of God's existence.
UPDATE (November 22, 2010): Also see the Internet Infidels' Atheist Web definition page. I now suspect that "empirical agnosticism" and "weak atheism" are indistinguishable.
UPDATE (November 24, 2011): Also see Jeff Lowder's January 4, 2006 post at Naturalistic Atheism, "Disagreement Among Self-Described Atheists about the Meaning of 'Atheism'" and Ted Drange's 1998 article at the Secular Web, "Atheism, Agnosticism, and Noncognitivism." Drange's distinctions seem to me to be well worth using. Maverick Philosopher's "Against Terminological Mischief: 'Negative Atheism' and 'Negative Nominalism'" is also good.
UPDATE (January 20, 2012): Jeff Lowder has written further on this subject at the Secular Outpost, in "The Definition of Atheism, the Anal-Retentive Defense of Etymological Purism, and Linguistic Relativism."