Weak atheism is usually confused with agnosticism.
Strong atheism
Strong atheism is the belief that there is no god. Usually, the strong atheists have some arguments that makes them believe that the god hypothesis is intrinsically impossible. A common counter-argument is that it is impossible to prove that there is no god. The counter-argument seems to rely on two assumptions: 1) That the strong atheist's only reason for rejecting the god hypothesis is that he cannot find any proof that theism is true and/or 2) that it is intrinsically impossible to know whether there is a god or not. The first premise is a confusion of strong atheism with weak atheism, and the second will be investigated further under popular agnosticism.
Sometimes the argument against atheism is a variant of the above and says that only because one has searched the whole world and didn't find a god, one can't draw the conclusion that there is no god. Usually the argument is presented as a parable: Suppose you search a forest for bears without finding any. Would that be proof that there is no bear in the forest? The answer is, of course, no. The bear may have moved around in the forest, avoiding you so you didn't see it, and even if there were no bear in the forest when you searched it, there may have moved some bears there after you left.
This is a misinterpretation of the atheist standpoint. Let me deny it with another parallel: A prime number is a natural number that cannot be divided by any other number than itself and 1 (one). Examples of prime numbers are 3, 5 and 57. Now, there can be no even prime number greater than 2 (two), and the reason is that all even numbers greater than 2 are divisible with 2. You don't have to search the whole infinite series of natural numbers to make that concousion. You can deny it on other grounds. For the same reason, even if the strong atheist is unable to search the whole universe and a possible supernatural world, there can be rational arguments that makes gods impossible, or at least implausible.
Another misinterpretation of strong atheism is that the strong atheist knows for certainty that there is no god. Some (if not all) strong atheists rather think that the existence of gods is just very unlikely. Atheism is not about knowledge, but about belief. The strong atheist simply believes that it is impossible for there to be a god.
All arguments for strong atheism are of the following kind (Modus Tollens):
If there is a god, then p is true
p is false (or p is probably false)
Therefore there is no god.
Of course, these arguments rest on the first assumption, that if there is a god, then p is true. All arguments have a premise that can't be proven from within the argument itself (Gödel's theorem). This is no ground for discarding strong atheist arguments though, because first of all the premise is taken from theism so if one disagrees with it, then one also disagrees with the theism that is disproved, and secondly, if this argument is discarded because of Gödel, then all arguments must be discarded for the same reason. You cannot eat the cake and keep it at the same time.
Weak AtheismThe weak atheist position does not need a justification - it is the default position. One should not accept a position unless there is some rational reason for supposing it true. For a weak atheist it is sufficient to say: "I don't know what a god is", or "I have never heard of a god". Unless theism can be proved in some way, the weak atheist position is the preferred position. This is often confused with agnosticism.
Agnosticism
The word agnosticism is derived from the Greek a- (negation) and -gnosis (knowledge). Simply put, an agnostic is one who says: "I don't know". One can be agnostic about many things, e.g. whether the moon is a cheese or not, but usually it is used about the existence of god. The popular definition of an agnostic is someone who doesn't know whether there is a god or not. This is different from the original definition, but first let me explain why this is not in any way opposed to atheism as defined above. Atheism is the belief that there is no god, or the lack of belief that there is a god. It has nothing much to do with what one knows, but more what one believes. It is fully possible to be ignorant of the fact whether there is a god or not, and still believe that there is no god, or not believing that there is a god, as this popular definition of agnosticism is not about belief, but of knowledge.
Sometimes agnosticism is popularly defined as one who doesn't believe that there is a god, while at the same time believing that there is a god. I don't think this "agnostic" position is possible. Either you believe that there is a god, or you don't. You can't believe that a position is true, at the same time believing that it isn't. Yet another definition of agnosticism is the belief that the existence of a god or gods is intrinsically unknowable. This will be addressed further down.
All popular definitions of agnosticism are unhistorical, and ignore the definition laid out by TH Huxley. Huxley was the person who coined the term in the 19th century, and he defined it thus:
"Agnosticism, in fact, is not a creed, but a method, the essence of which lies in the rigorous application of a single principle. That principle is of great antiquity; it is as old as Socrates; as old as the writer who said, "Try all things, hold fast by that which is good" it is the foundation of the Reformation, which simply illustrated the axiom that every man should be able to give a reason for the faith that is in him; it is the great principle of Descartes; it is the fundamental axiom of modern science. Positively the principle may be expressed In matters of the intellect, follow your reason as far as it will take you, without regard to any other consideration. And negatively In matters of the intellect do not pretend that conclusions are certain which are not demonstrated or demonstrable. That I take to be the agnostic faith, which if a man keep whole and undefiled, he shall not be ashamed to look the universe in the face, whatever the future may have in store for him." (TH Huxley, Agnosticism, 1889)
As we can see, there is no contradiction between Huxley's definition of agnosticism and any of the atheist positions; it is even compatible with theism. As long as the conclusion can be demonstrated, it is in line with agnosticism. Most people who hold the position of Huxley would probably be classified as weak atheists according to my definition above. Many would call themselves skeptics. Note also that Huxley does not explicitly speak about God or gods, but rather of knowledge in general terms. (Other quotes by Huxley show that he is talking not just about gods, but about the ultimate questions in general.)
Popular Agnosticism
Most modern dictionaries seem to ignore Huxley's definition. Instead they define agnosticism as the belief that god is unknown and/or unknowable, but if god is unknown the weak atheist position is the default position. I call the dictionary version of agnosticism "poplar agnosticism".
As far as I can see, popular agnosticism is self-contradictory. If god is unknowable, how is it that the agnostic knows this fact? By claiming that the gods' existence is unknowable, the popular agnostics contradict themselves. If no knowledge about the gods is possible, then it is not possible to know that fact. It may be possible that the popular agnostic doesn't know whether there is a god or not, but that is no proof that the atheist does not know. It is a logical fallacy to impose your own ignorance on others. Many self-acclaimed agnostics bash theists and atheists for claiming knowledge about the unknown, but what about themselves? They too claim knowledge about the unknown. The only difference is that the popular agnostic claims that the unknown is also unknowable. Both the theist and the (strong) atheist claim that it is knowable and that they know. What if they are right? How is it possible for the agnostic to bash the theist or the atheist for knowing something the agnostic does not? It may be that the other have some good arguments that the agnostic is unaware of.
Conclusion
I have defined atheism using the two definitions of denial. An atheist can deny, either that there are gods (strong atheism), or that there is a ground for theism (weak atheism). The popular definition of agnostic is unhistorical. The original definition of agnosticism is not in contradiction neither with theism nor atheism. The popular definition of agnosticism is illogical and sometimes contradictive.
When it comes to belief about the gods, there are three possible standpoints:
The belief that there is a god (theism)
The lack of belief that there is a god (weak atheism)
The belief that there is no god (strong atheism)