They are replies, but I think that is another word for posts.
Categories have a "c" in their URL: http://discuss.codecademy.com/c/Meta
Topics have a "t" in their URL: http://discuss.codecademy.com/t/badges-howto/7283
Replies are Posts, they add an incremental integer to the end of the "t" URL that they live in: http://discuss.codecademy.com/t/badges-howto/7283/9
From the Badges page: "Read every post in a topic with more than 100 posts".
To get the Reader badge we need to have a topic with 100+ posts in it. Our forums are new, and we don't have a topic that meets that criteria yet.
Last night I went to another Discourse site, one that did have some topics with 100+ posts and read one of them ... it took a while I don't know if it can distinguish one just scrolling through them all quickly or does it look at the time it takes you, but I went slow just in case. When I went back to check today, I found that I had the Reader badge.