What do you see when you press the power button?
If absolutely nothing happens then unplug the PC from wall, wait 13 seconds minimum then plug back in. This resets the part of the PC that looks for the PC button being pressed. There is a small chance the PC will now power on.
If you see something happening on the screen, or you hear fans starting to whirl when you try to power up please describe all the things you see and hear. This helps understand which part likely failed.
Sometimes the pc power button itself fails. Try pressing it at different edges. Let us know how much work you are willing to do inside the PC case, the switch is bypassable.
If you decide the PC is dead (that means you do not want to fix it): The data stored on a PC is in the "hard drive". That hard drive can be removed from a dead PC and read from a new PC fairly easily. If you have pictures, etc on the HP they are on the hard drive. If you want to keep them then do not throw the PC away. We can give you instructions on how to move the drive to a new PC or laptop. (Basically you pull the drive out and put it in a $20 USB attached case then read the drive on your new PC just like you read any other USB flash drive, but you'll likely want the step by step to do this)
Can you post the PC model number? On hp pavilions it can be on a sticker on the bottom or lower back of the PC, sometimes on a sticker on the top near the back. It'll say something like " hp pavilion E9280T" and be near the serial number.