Sounds like an IP conflict. Your router is giving your Wii the same IP address as the Computer.
Do you maybe have someone close to you that knows anything of Routers? Cause you may have to ask that person to check the IP settings on your router. (Ofcourse, that could be explained here, but every router has different settings and stuff so..)

EDIT:
Or maybe you can check the wii or computer for a static IP address. You can check this in the WiFi settings of your Wii.
For your computer tho, go to:
Start > Control Panel > (and now it gets tricky since I dont know you windows version) > Network Centre > Network Adapters.

Anyway, get to your network adapters.

Then right click the one you use to connect to the internet, click on properties, then double click on Internet Protocol Version 4, and make sure there is no static IP on your PC, or maybe change it so you won't get any conflicts.

 

It's only one of the computer, and it says nowhere that the PC is wireless.
It sounds to me that, if the wii is turned on, it gets an IP from the router, and all the other PC's in the house also get one, but just not that 1 computer.
And if its turned off, the IP conflict isn't there, so the PC gets internet again.

You my guess is that your guess is a bit far fetched.

 

Usually at least 3 computers are connected wirelessly at a time.
Sometimes more when I connect a phone or the Wii..

 

Had the same issue the other day, IP conflict.

Just reset the client network connection that has problems.
Make sure all clients are logged on the network as well.
The router will give the new IP addresses.

If you Wii is connected via Static IP, MAC or virtual server, it's IP will not be given away twice.
Make sure you don't have the Wii connected via a static IP and DMZ forwarding at the same time.
This can cause conflicts.

If you have conflicts without the Static IP/virtual server setup, I suggest you to so.
There could could be a problem how you router gives the IP's away dynamiccly. ( DHCP )

 

I guess youll need another router... everything is wifi these days

 

well it could be a few things, firstly check to see if any port forwarding is set to the wii, also something like dedicated IP or something like that can cut off all other clients to the router, there is another thing aswell and that is called DMZ same as the first thing, you can always assign the wii a static IP as stated by other people which is quite easy to do, if your router has wifi then unless you have more than 255 devices connecting to you router then you could come across an IP conflict.

if you dont know how to set a static IP in your wii follow these steps, if you do then disregard.

1) if your router IP address is is for example 192.168.1.1 then assign the wii an IP of 192.168.1.200 up to 250
2) subnet mask if you have a home account then it wil be 255.255.255.0
3) default gateway will be the address of your router IP


make sure you have WPA2 encryption enabled so every tom dick and harry doesnt connect to your network.

sorry if I have been a bit vague in parts I am bunged up with a head cold, hope you fix your problem soon.

 

I'm going to hazard a guess and say that the OP didnt understand that..... atleast I didn't.....

 

Ya, Am0s has it about right.
I could of said NAT:ed instead of "un-bridged" but that would just involve another tech word that most don't understand ;P

And yeah, they could do it the subnet mask way of assigning a couple of IP's but for all the ISP's i've had experience with in Sweden it's done in some other way, some ISP's giving out 3 public IP's ( that would be an 255.255.255.252 /30 subnet btw, but then you don't count the network address and broadcast, which would only leave 2 public IP's...) and some are giving out 5 public IP's ( /29 network, which would waste at least 1 IP for each customer..) and so on.
So i don't think they do it that simple way, also the public IP's I've seen on the same account have not been in order - which makes this whole thing not working that way (at least in my cases) confirmed.