What to do with my iphone that won't turn on?

Anytime my iphone 4 is plugged into the wall, it turns on, but as soon as you unplug it, it shuts off. It doesn't even charge when it's plugged into the computer. What should I do, how can I fix it?

Seems like your battery is most likely done, and the power coming from the wall is enough to turn it on, while your computer doesn't put out enough power to turn it on.
Just take it to an apple store, they'll take care of you.