In Html5 you can use the object tag to include a file.
Basically you create a single file containing your header and the common code that goes on each page.
Then on every page of your site you add
<object name="includedfile" type="text/html" data="page.inc"/>
where you need the content to appear.
Edit:
Check also jquery if you prefer to use javascript. There are easy functions to achieve the same result like:
$.get('test.html')
.success(function(data) {
$('div.content').html(data);
});
Where test.html is the page that you want to load and div.content is the place where you want to put the loaded code.