Z-index isn't the issue here. It's improper use of position. position:relative, by itself, does nothing other than allow you to position something (with top/right/bottom/left) relative to where it would have naturally appeared. in your case, the twitter div is the first element, so it is appearing at the top of the page.
I gather from your post that you want it just to the left of the slideshow div. In that case, move the tweets div as child of #container. Give #container position:relative; and position:absolute; to #tweets. This means that when you set a position with top/left or top/right #tweets will move relative to #container. In this case, top should probably be 211px and either left:-389px or right:900px.
EDIT: It seems you're fiddling around on the file you linked to so I can't be sure what you're after as now you've applied position:fixed which, as the name implies, fixes the position of an element relative to the browser window and not the document.
Sorry for fiddling Frinkky.
Yeah. I decided I prefer the Twitter feed on the left edge away from the other content.
I really appreciated your thorough explanation as I have struggled on other projects with z-index.