Do you need two ip addresses to connect both your laptop and wii to the internet at the same time

If you have your modem connected to a switch and want them both connected to the Internet at the same time then yes. Two same IP addresses on the same network equals major trouble. To do what you want to do I'd suggest buying a router. A wired one will work if all you need is wired access. The router will host your Internet connection to all of the computers connected to that router including ones that are connected to a switch branching off that router. This is how it works:

1. You make a request for the from a computer that is part of your LAN.
2. The request has the LAN IP address (private IP address) of that machine on it.
3. The router takes that request and changes the private IP to the Internet IP of the router.
4. The router keeps track or what machine on the LAN requested this request.
5. The request goes out to the Internet.
6. The server that you requested sends the information your requested back to the router.
7. The router looks in it's log to see what LAN IP to forward the request back to.
8. The router changes the WAN IP of the information back to the LAN IP of the machine.
9. The information gets sent to that machine.

The process is known as NAT (Network Address Translation)

Hope this helps you out!