Try the following and see if it will turn on.
1. First make sure the device has been charged.
2. Unplug your charger cord.
3. Remove the battery.
4. Press and hold the "Power" button for 30 seconds and see if the device will turn on.
5. If it does, great, if not yet, no big deal. Now plug back in the power/charger cord. (If it didn't turn on without the cord in, it should turn on once you it is in. Yes sometimes they will turn on, or try to turn on, even with nothing attached/plugged in.)
6. Now put the battery back in.
Should that not be the problem, then check the battery. Try running the laptop without it (some will, some won't) and if that doesn't work, then you may need to replace it.
NOTE: If you got an aftermarket cord, or choose to get an aftermarket battery, this can also be the problem. While aftermarket is great for many things, laptop charger cords and laptop batteries are not one of them. It is common for aftermarket cords to not put out enough power, and the same for batteries.