Integrating certain WordPress elements into existing site

Hi, I haven’t used wordpress since version 1 and I would like to know if I can do the following with the newest version. I have an existing static website with HTML/CSS pages and I want to know how I can have the posts be integrated into one of these pages. Essentially what I want is to have the posts show up on my HTML page and the categories selector on the right hand side and that’s it. If a user was to select posts for another category, those posts would appear in the same manner, under the same html layout, within the same divs etc.. How can I establish this? Help is much appreciated.