Web Page Controlled Arduino Over Wifi

I saw similar questions but tried to put them together with my questions.
I want to make wifi controlled arduino to turn led on.From anywhere in the world using a webserver ,let say www.arduino.com.But i dont want to dependent on router or plugging arduino with usb to pc and controlling it from just that pc.
Therefore i want to learn what should i know before going for it.. Is it possible?

When i search it i saw some solutions for wifi controlling;
Arduino Wifi Shield and CC3000 Wifi chip.

There are other names such as Xbee and Wifly but i am not sure about them.

I want to understand first of all, if i want to control arduino from a web site over wifi, what kind of part i should use..Any name?
And when i found it..Can i use it from anywhere .(not just from local
host) From my own web page on the net?

Also lastly, i see on the net when they always use web page examples its all about about php and i am confident with asp.net and ok i will make a simple page but can i do it with asp.net?
What other things should i know?