HTTP is the Hyper-Text Transfer Protocol;
It is the internet protocol used to transfer web pages.

A Web Server is a system that makes web pages available to Clients;
Probably the most familiar Web Client is the Browser.

A Web Server requires an HTTP server to transport its pages, and a Web Client (eg, a Browser) requires an HTTP Client.
Because of this, it can be quite common to hear "web server" and "http server" used as if they were equivalent - but they're not. (I might have been guilty of that in this thread).

Diagrammatically, it's something like:

   +-----------+                          +-----------+
   |Web Server |                          |Web Client |
   +-----+-----+                          +-----+-----+
         |                                      |
         |                                      |
   +-----+-----+                          +-----+-----+
   |HTTP Server|                          |HTTP Client|
   +-----+-----+                          +-----+-----+
         |                                      |
         |                                      |
         +---------------Internet---------------+

Or this:

   +------+-------+                           +------+-------+
   |      |       |                           |      |       |
   | Web  | HTTP  |                           | HTTP | Web   |
   |Server|Server +<---------Internet-------->|Client|Client |
   |      |       |                           |      |       |
   |      |       |                           |      |       |
   +------+-------+                           +------+-------+

So, to give your embedded device a Web-Server interface requires two things:

The "web server" part that creates web "pages";

The "HTTP server" part that responds to HTTP requests.

You can make the "web server" part as simple or as complex as you like: it doesn't have to produce fancy graphics, animations, etc - it could simply provide plain text!

At its simplest, the server simply needs to fulfil GET and POST requests from the client:

GET is a simple request from the client;

POST allows the client to provide some data in its request.

Again, see this tutorial:
http://www.jmarshall.com/easy/http/

Of course, this all still begs the question of whether the webserver is an appropriate solution;
Maybe you could make your devices web clients instead;
Maybe HTTP isn't necessary or appropriate at all...