My iPhone won't turn on
How exactly are you attempting to charge the phone? If the phone is completely dead, you should not use a USB port on a computer or a USB hub to charge it.
Instead, I recommend that you obtain the cube-shaped Apple adapter that plugs into an AC outlet if you don't already have one. It's just a little white cube that plugs into the wall, and it has a USB connector on it. Use an iPhone-to-USB cable to get the phone charging from the wall using that adapter.
Leave the phone connected this way for at least 15 minutes. When the phone gets to a certain minimum level of charge, it should display the battery image on the screen for a few seconds, to let you know it is now beginning to recharge. Note that it may take several minutes before you ever see anything on the screen. Only then will you know it's actually charging. Leave it for a while, and let it charge. If the sound is turned up on the phone, you will also hear a beep when it displays the battery image for the first time.
You can check status periodically by briefly pressing either the round button near the bottom of the phone's face, or the power button on top. It will display the battery image, which will reflect the current state of charge.
Once the phone has at least 25% or more charge on it, you should be able to power it up (while still connected to the charging source). To do so, hold down the power button on the top right edge of the phone until it turns on.
If the phone does not begin to respond after 15 minutes of connection to a known-good charging source, then it has developed a hardware problem and will need to be repaired or replaced.
I can't stress enough that you need to make sure the problem is not the manner in which you are charging it, or the adapter or cable that you are using to charge it. Try multiple cables if you have them, and obtain more than one of the AC adapters to make sure that isn't the problem. If you are using a charging dock of some sort to charge it, that could be the problem as well. You want to eliminate all other possibilities in order to determine for sure that it's the phone.
The only other weird situation that I've seen is if you have a corrupt app that is blanking the screen, making it appear as though the phone is off. To make sure this isn't the problem, press and HOLD the power button on the top right edge of the phone for at least 10 seconds or more. You should see a message that says "slide to power off". Slide the button to power off the phone. Wait until it goes completely black, then wait another 10 seconds. Then press and HOLD the power button on the top right edge of the phone again to restart it.
Cheers,
-=Cameron