Computers often fail to boot due to faulty components, such as a broken power supply or hard drive. Driver errors, firmware problems and computer viruses can also cause the operating system to fail to load.
If a computer does not power up at all, with no fan noise or power lights, then there is probably a power issue. Check that the computer is plugged in, that the power sockets used are working, and that all the power switches are set correctly. If the machine still fails to power on, then the power supply could be broken. If the computer is a laptop, then the charging cable could also be faulty.
If a computer powers up, but does not load the operating system, then it can be due to a wide variety of hardware or software issues. Hard drive failure is a common reason.
TechRadar.com and PCAdvisor.co.uk have detailed guides for identifying and troubleshooting these issues, but the guides are not comprehensive. If in doubt, check your computer warranty and seek technical support.