HTTP is an abbreviation for HyperText Transfer Protocol and HTTPS is an abbreviation for HyperText Transfer Protocol Secure .
What do they do?
HTTP and HTTPS are basically doing the same thing, HTTPS is just a secure/encrypted connection. It is the part of World Wide Web that makes links work, so that you are able to click your way through pages. When you click on a link, there has to be something that transform the click into a command about going to the right page and open it. Today this seems obvious, but when all of the World Wide Web was invented, this was ground breaking technology, and the reason the Internet became accessible for John Q. Public.
HTTPS is also found in another variant called S-HTTP (Secure Hypertext Transfer Protocol), which isn't used much, and according so some sources is being discontinued.
Like many other things in the WWW technology, this is all running in the background and is more or less invisible for the common user. These days you can usually tell from the browser's address whether you are on an encrypted page or not, because if secured, it is specified as https://sitename, whereas many browsers operate with http:// as being implied, if it doesn't say https://, and thus something that can be hidden/omitted.
When working on a site, you work with a set of files on a drive, and not directly in the files that have been uploadet to the web server (unless working with a content management system (CMS)) When opening this file in the browser, you will see that instead of http://, it says file:/// before path and file name. It is still http://, and links still work. For some reason it has been chosen that, when displaying HTML files from somewhere other than a web server, it has to say file:///.