Your best methods are PHP files with includes. SSI works well, but you have to have the .shtml file.
Non recommended methods would be using an iframe to include the header. Javascript can also be used but it isn't good to use client side includes. jQuery would also be client side includes.
You can set .html files to use SSI but if you have restrictions this is most likely a restriction that you can not get past. Hosted server accounts do not allow this for the most part.
Apache Server docs on SSI. http://httpd.apache.org/docs/current/howto/ssi.html
To permit SSI on your server, you must have the following directive either in your httpd.conf file, or in a .htaccess file:
Options +Includes
This tells Apache that you want to permit files to be parsed for SSI directives. Note that most configurations contain multiple Options directives that can override each other. You will probably need to apply the Options to the specific directory where you want SSI enabled in order to assure that it gets evaluated last.
Not just any file is parsed for SSI directives. You have to tell Apache which files should be parsed. There are two ways to do this. You can tell Apache to parse any file with a particular file extension, such as .shtml, with the following directives:
AddType text/html .shtml
AddOutputFilter INCLUDES .shtml
One disadvantage to this approach is that if you wanted to add SSI directives to an existing page, you would have to change the name of that page, and all links to that page, in order to give it a .shtml extension, so that those directives would be executed.
The other method is to use the XBitHack directive:
XBitHack on
XBitHack tells Apache to parse files for SSI directives if they have the execute bit set. So, to add SSI directives to an existing page, rather than having to change the file name, you would just need to make the file executable using chmod.
chmod +x pagename.html
A brief comment about what not to do. You'll occasionally see people recommending that you just tell Apache to parse all .html files for SSI, so that you don't have to mess with .shtml file names. These folks have perhaps not heard about XBitHack. The thing to keep in mind is that, by doing this, you're requiring that Apache read through every single file that it sends out to clients, even if they don't contain any SSI directives. This can slow things down quite a bit, and is not a good idea.