When you are connected to the Internet, you actually have two different IP addresses, a private LAN IP and a public WAN IP.
In most home network applications the router connects your local group of computers and devices known as the LAN with the Internet, commonly called the WAN or Wide Area Network.
The router usually assigns unique local IP addresses to all of the devices connected to it via a service known as DHCP. The addresses assigned by your router are private addresses http://en.wikipedia.org/wiki/Private_net... and are not routable across the Internet.
Your cable or DSL modem will get a Public WAN IP address from your ISP. These public IP addresses are leased in blocks by your ISP and are Internet routeble. The DSL or cable modem is connected to the WAN port on your router. The router will manage the traffic between the devices on the LAN and the Internet.
The only IP address that a website or outside device can see is your public WAN IP. They can't see the private IP addresses on the LAN side of the router.
You can see your current public WAN IP address here: http://www.mywanip.com/
Depending on your ISP your public WAN IP address might be dynamic (changing periodically) or static (fixed).
If you have a static WAN IP, the only way to change it would be to call your ISP. However if you have a dynamic WAN IP you can usually get your ISP to generate a new one by unplugging your broadband modem for a few minutes and then restoring power. Just cycle power, do not press the reset button.
Since your WAN IP comes from your ISP, and is required for you to communicate with the Internet, then there is no simple solution.
The most common way of hiding a WAN IP address is to use a web based proxy server. Your HTTP requests are sent to the proxy server. The proxy server forwards your request to the destination server, and then fetches the requested content and relays it back to you.
Proxy servers have risks, especially free ones where you don't know where your data goes or who has access to it. http://whatismyipaddress.com/proxy-risks
There is no way to really hide your WAN or public IP entirely, otherwise no data could be sent to your computer from the Internet as it would not know the destination address.
Good Luck..