Note: Before doing anything listed here, you should take it to the place you bought it, or contact your carrier or manufacturer if it is possibly still under warranty. Many of the things listed here would make a warranty null and void, so check with them first.
There can be (on some devices) a small pin hole that will reset it.
However, have you tried holding down the power button for around 20-25 seconds? Also, are you sure it is fully charged? If it isn't fully charged it may not turn on.
If this doesn't do it, the problem may be the button, the charger cord, the charger port, or the battery.
Make sure the power button isn't stuck and it is actually moving the internal button ( the outer 'button' is just a cover to the internal one).
Also make sure your power cord is actually charging the device. If it isn't, check to make sure that there is nothing in the port or cord end that could be restricting it (dust, dirt, etc.,). If there is nothing in it, then you want to make sure the cord is actually good. Either by getting a different cord to plug your device in with, or using your cord on a device you know is working to see if it charges it. Should this show your cord isn't working, you would need to replace it.
Should you find your cord is working fine, but not working on your tablet, then you would need to determine if it is the port that isn't working or the battery. Either one would require the tablet be opened up. For the port to be replaced would usually require the motherboard to be replaced. For the battery, you just need to buy a new one and install it.
If you don't feel comfortable opening it up and doing these yourself, then I would suggest going through a local tech.