I search for this constantly, and although many of these answers are quite right in citing that V8 isn't iOS compatible, it seems that running a nodelike environment within iOS isn't all that impossible.

The first option I discovered was called "Neunode" (hasn't been updated in a year). Here are some reference links if you want more:

• https://github.com/snakajima/neunode
• http://www.neunode.com
• https://www.youtube.com/watch?v=xzFH80-HYTo (httpserver example)

Nodelike / Node.app Project

Nodelike is the core framework of the Node.app project, which has the goal to implement a roughly Node.JS-compatible interface using JavaScriptCore.framework on iOS 7 and OS X Mavericks.

Example: Interpreter

an example project implementing a Node.JS interpreter as an iOS app, utilising the Nodelike framework.

Edit:

It is still "not viable for serious projects" but checkout updates at:https://github.com/node-app/Nodelike

Overall, The Node.app Project looks promising, be sure to checkout their website

Yes! You can run Node applications on iOS devices. Node.JS fork JXcore has implemented SpiderMonkey JavaScript engine in addition to Google's V8. The source code repository is available from Github

No. But you can write native applications in Javascript (which is the same language as Node - which is server run javascript) by using a tool like Unity.

Also, welcome to SO!