I'd like to expand on volkerjaan's answer. I am Asperger myself. I know what :-)
, ;-)
and ;-(
mean (though I know ^^
only as see above). I don't necessarily understand when people are using them. I do understand their origins, and I suspect that is what the question boils down to.
To explain what they mean:
:-)
— person feels happy
:-(
— person feels sad
;-)
— person is joking
To follow the conversation, it is not needed to understand the meaning. It is enough to know the meaning. In my interpretation, understanding is related to why. As an Asperger, the ability for abstraction is often different from those in neurotypicals. So I would explain it as follows.
Emoticon explanation for Asperger
When we draw a face, we like to simplify things. For example, sometimes we draw only a circle with dots for the eyes, and a line for the mouth. For example, from the Wikipedia article on smiley:

This represents a face in an abstract way. The face is happy, because the edges of the mouth are turned upward. Of course, a real face looks very different. However, this is an abstract representation of a happy face. For an unhappy face, the edges of the mouth would go down.
When we are on IRC, or in another text-only medium, we do not use pictures. Therefore, we need to make drawings in a different way. One way to make drawings on IRC is by using characters normally used for other things. This is what we do when we use emoticons.
A emoticon is a sideways drawing of a face. To see it in the correct orientation, turn your head to the left, so that you see the :
or the ;
on top, the -
in the middle and the )
or (
at the bottom.
In :-)
, when turned side-ways, we see three parts: first the :
, then the -
, and finally the )
.
The :
represents the eyes. They are two black dots (depending on your viewer) like in the picture of the smiley below.
The -
represents the nose. The nose is not shown in the smiley before. Sometimes, it is not shown in emoticons either, and people write :)
instead of :-)
. People are lazy~
Finally, the )
represents the mouth. Like in the smiley above, the edges of the mouth are turned upward. This means happy.
For :-(
, the first two parts are the same: :
are the eyes and -
is the nose. The third character is different:
- The
(
in this smiley represents again the mouth, but now the edges of the mouth are turned downward. This is a representation of being sad.
The third emoticon is ;-)
. It is again similar to the first one, :-)
, but now the first character is different.
- Remember how
:
represented the eyes. In this case, ;
represents the eyes, but with a wink. The ;
consists of something that looks like .
and something that looks like ,
. The .
is the open eye and the ,
represents the closed eye. This is strange: why is the closed eye larger than the open eye? It should be the other way around! Yes, it should. Even if it is wrong, people use ;
when they mean: two eyes, one of the two is closed. When one of the two eyes is closed, it is a wink.
People use the wink when they mean I am joking.
There are many other emoticons, but those are the most used ones. There is a list on Wikipedia. This lists the meanings, but it does not explain any of them. I don't know if there is a website anywhere with explanations. Sometimes, you just need to accept what it means, without understanding them. But I hope that my description above explains the most common ones.