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.
So, your LAN connected devices will all have unique private IP addresses assigned by the router and share the WAN IP connection via the router. No one else can have the same WAN IP as you at the same time or the Internet servers would not know where to send the data packets.
The only IP address that a website or outside device can see is your public WAN IP.
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.