An ID means there is only one on the page, there can only ever possibly be that one on the page, and it should probably represent something specific and special.
A class means you want to apply some CSS rules or Javascript processing to it.
<div class="container">
<div id="page-navigation" class="left"></div>
<div id="page-content" class="center"></div>
<div id="page-sidebar" class="right"></div>
</div>
The first DIV is the one, single container used by the page for the main content. It doesn't need an ID because you don't need to do anything special to it. It does act like a generic .container.
#page-navigation has links or whatever relevant to the page. It has an ID because you might want to style the contents differently. Inside the container it acts like a .left.
#page-content is the content for the page. It has an ID because maybe it has some special border or background, or maybe you want to run some Javascript on what's inside. It acts like a .center.
#page-sidebar has maybe ads or additional information or something like that. It has an ID because that way to can target it with Javascript to place the ads. It acts like a .right.