The matter is rather the other way around. What is called "The Elder Scrolls" is the game series. "The Elder Scrolls V: Skyrim" is one of the games belonging to that series.

As for other versions of Skyrim, there exist three official expansion packs under the names Dawnguard, Hearthfire, and Dragonborn, but these are merely additional content and not stand-alone games.

You can read more here: http://elderscrolls.wikia.com/wiki/The_Elder_Scrolls

Morrowind and Skyrim are examples of provinces in the continent of tamriel. The Elder Scrolls is the title of the series of video games, named after the in-game objects, which take place in Tamriel. Each game in this particular series (save the first game and excluding spin-offs) (at least so far) takes place in a different province, and the game is usually named after its province (TES V: Skyrim takes place in Skyrim). That being said, there is only one province of Skyrim and there is only one game entitled "Skyrim", but there are more games in the game series and there more provinces in tamriel.

EDIT: Looking at it now, you seem to have the names switched: there are other Elder Scrolls Games, but you have Skyrim.

To clarify it. 

The Elder Scrolls 1: Arena

The Elder Scrolls 2: Daggerfall

The Elder Scrolls 3: Morrowind

The Elder Scrolls 4: Oblivion

The Elder Scrolls 5: Skyrim

The Elder Scrolls: Online

Ingame Skyrim is one Province on the Continent of Tamriel on the planet nirn. Just like New York is one State of the USA on America.

But I've heard of others called Oblivion and Morrowwind...