I know this is obvious, have you checked that your mains power is actually working and switched on. tried something else in the same socket. The same goes for the computer, have you tried a different usb device in the same port?
USB cabling is very simple. it has Voltage+ line (5V 1A), Voltage- line (0V ground) Data+ and Data-. This is univeral across all USB cables and any adapter or socket with the correct size USB connector for the phone should work. There's also not much of a way it can go wrong beyond a broken wire.