Webserver on Arduino
You'll need something like the Ethernet Shield to connect your Arduino Uno to a LAN.

As for webserver functionality, the Webserver Example sketch does the bulk of what you need.



For the "access this data through the internet" part, assuming you want access beyond your LAN; you'll need to perform Port Forwarding on your local router (i.e. allow the "Internet" to see your Arduino).

If you need wifi, then personally I prefer the CC3300 to the Arduino Wifi shield.