No Display
If you are able to turn on your Wii but can't see anything on the screen, ensure that your Wii's input and output cords are connected properly. The Wii can use both regular composite as well as 480p-compatible component cables, so it is important to ensure that your inputs match with your outputs, and that you are not connecting a composite input to a composite output. Also make sure that your television is tuned to the correct input channel. Most modern sets have several input channels, so make sure that you have cycled through all the possible input channels before panicking.
Power Issues
If your Wii will not power on or the power doesn't stay on for very long, then you may have a problem with your AC adapter. If you have had a recent thunderstorm and do not have a surge protector, the adapter may have blown. Power problems with the Wii can also be caused by unlicensed peripherals that are connected to the console. If your peripherals do not have a "licensed by Nintendo" seal, they are probably unauthorized and could be sucking too much power away from the console, which would result in it unexpectedly shutting down.
Wii-mote Not Responding
If your Wii-mote doesn't seem to be working the way it should, there are a few things you can try to fix it. Ensure that the sensor bar is correctly connected to the back of the console. Also, make sure that the sensor bar is properly placed in the center of the television, preferably on the top of the TV screen.