• The BIOS determines if the video card or onboard video is working.
• The BIOS verifies RAM.
• It checks for a keyboard and mouse.
• It looks for a PCI bus and then checks the cards that are installed.
• The system information is then displayed on the screen.

• Remove all power (adapter power Cord and Battery (for laptops))
• Hold down the ON button for 30secs (dispersing internal voltage)
• Plug the power cord in only (ie leave battery out, for the moment)
• Turn on

Many No POST failures are caused by your computer's CMOS settings being corrupt
Resetting the CMOS to defaults will cure this, and possibly allow your computer to POST
By reading your computer's or motherboard's Manual (discussed further down this page)
Locate the CMOS battery and/or the "reset CMOS jumper" on your motherboard
Following the reset CMOS procedure (usually shorting two pins, with CMOS battery out)
Your CMOS settings will be reset, hopefully allowing your computer to startup (or POST)

Firstly it's always best to download your computer, or motherboard, manual
You can locate your manual, by going to you manufacturer's support page
For convenience (and as an example) Here is Dell's and HP's support info
Make sure to follow all AntiStatic and power recommendations/precautions

Note: Although this subject has been discussed in the "Benchtesting" link above,
Just because some lights come on, does not mean your Power Supply is working properly.

It is also possible that your actual power switch is faulty
Or even the Display off micro button is faulty, (on laptops)

In most cases, it's a matter of testing with a known working Power Supply
Here's an excellent link on Power Supply information and troubleshooting