The HTML files are the webpages themselves. The website requires 2, possibly 3 components
1 - webpage
(done)
2 - server
to host the site (free ones usually have ads i'm afraid)
some web hosting is remarkably cheap anyway - some providers do it for less than £10 a year, if you want a site I'd just bite the bullet - at the end of the day, even if it doesn't have high commercial value, you will have learnt a bit about web development and if you go with option 3 (below) then you will have somewhere to have some decent - company branded - email addresses.
3 - domain name,
ie - mybusiness.com - optional, but at todays prices - recommended
123-reg is a popular one, with .co.uk domains for less than £10 a year
Google "web hosting" and "domain name registration" and you will find a lot of companies out there wanting to sell to you! it is a very saturated market!
Additionally, when you have access to a server to host your website, you will be given FTP access to that server - as benjamin_c mentioned, it is a method of sending files to a web server - you can use the FTP protocol to send your html pages to the server.
In this case, I'd recommend spending a little bit of money on it and having a good try.