Computer won't turn on (Laptop)

It may be the laptop's battery that prevents your laptop from turning on. Try turning your laptop on directly through the AC adapter (remove the battery first). If it does turn on, then it is the battery itself that needs replacement.

Note: Buy a new and unused battery only. They usually last for two to three years (depending on the manufacturer). Also, it may last longer if you use it only at places where you don't have access to electricity, or if you are at home, you can connect your laptop to the adapter and remove the battery. 

Maybe one of the external devices connected to your computer has malfunctioned. Remove any CDs, DVDs, USBs and other external devices (such as portable HDDs) that are connected to the computer. Then turn the computer on and check if it starts normally.

If it does, start returning the external USB devices one by one so that you find the faulty one.

Maybe there is an issue with your battery. Remove it, and start the laptop with only the power cable plugged in. 

Mabye your computer does turn on, but there are issues with your screen/monitor. Check out the Screen is blank solutions. 

Maybe the power on/off button has malfunctioned and that is why your computer doesn't turn on. Replacing the power on/off button will fix the issue.