While MoinMoin currently has no full support for HTML templates, and switching between several layouts, you can change the most important parts of the generated HTML code, and add your own HTML fragments.
"title1" and "title2" are inserted into the output right before and after the system's title html code (title1 is right after the <body> tag and normally undefined, title2 defaults to the "<hr>" above the page contents). "navi_bar" is a list of page names1 that are added to the title area as quick links into the wiki; if it is None, no navigation bar is added.
Similarly, the footer area can be changed by setting "page_footer1" and "page_footer2".
"html_head" is added into the <head> element for all pages, while "html_head_queries" is sent only for edit and action pages (html_head is sent for those, too).
See http://openlook.org/ for a very nice example.
1 In versions prior to 0.11, navi_bar was a piece of HTML code.