No Signal, or monitor going into Sleep Mode, or Power Saver Mode.
No video signal is coming from the computer to the monitor.
1) Check the monitor cable. Substitute with another KNOWN to be good monitor cable, or if it is attached use a KNOWN to be working monitor, and cable.
2) Monitor cable checks out OK the problem is the computer.
The two leading causes of desktop computer failure is the computer is dirty inside, and Power Supply failure.
FOLLOW Anti-Static Precautions;
Computer on a table, computer unplugged from power, computer case open. TOUCH an unpainted surface, of the metal frame of the open computer case.
This action will relieve your body of Static electricity.
IF you leave your computer in the middle of working on it, be SURE to Touch the metal frame again upon your return.
Use a can, or two, of compressed air for computers, and clean out the inside of the computer. Attach the plastic straw provided into the nozzle, and break the plastic lock tab off the top of the nozzle.
Pay special attention to the Processor fan, and finned Heatsink under it.
(You may need to use Q-tips also, to 'break the crust' of the Gunk, so the air can remove it)
3) Computer isn't dirty inside, or cleaning brings no joy?
Power Supply could have a weak voltage power rail.
(There are three; 3.3 Volts, 5 Volts, and 12 Volts)
1) If ALL of the LED lights were on at once, they would use less than 1 Watt of power
2) EACH fan uses 2 to 3 Watts of power.
3) A typical Processor can use 51 to 125 Watts of power. Just depends on what Processor it is.
This is why you may see LED lights light, and maybe fans spin, but the computer doesn't come on.
NOT enough power to turn the Processor on.
Suggest test the 3 main voltages coming out of the Power Supply, or use a KNOWN to be good, Compatible power supply for a test unit.
Perhaps there is a working unused computer you can borrow it's Power Supply, for a test unit.
An economical multimeter can be purchased for as little as $5 to $12. Available in a multitude of stores. An auto parts store is but one example.
If you know how to use a multimeter set it to DC Voltage. (Unless it is one of the ones, that automatically sense whether it is AC voltage, or DC voltage)
Use straightened out paper clips inserted into the back of the 20-pin, or 24-pin ATX main power cable's connector,http://www.playtool.com/pages/psuconnectors/connectors.html#atxmain24
Main power cable is plugged in, as shown in the photo to the far right.
The straightened out paper clip goes down into the socket hole, RIGHT NEXT TO the existing wire in the socket hole.
Has to slide down in the socket hole past the insulation of the wire, and touch a metal terminal at the end of the wire.
Orange wires are 3.3 Volts (DC)
Red wires are 5 Volts (DC)
Yellow wires are 12 Volts (DC)
ALL Black wires are Ground wires. (Negative)
Power supply turned on; (Computer turned on),
Paper clip in Orange wire socket hole. Positive (Red) probe lead touches it.
Paper clip in ANY socket hole with a Black wire. Negative (Black) probe lead of multimeter touches it.
You should be reading VERY close to 3.3 Volts (DC)
Do the same for a Red wire, (5 Volts), and a Yellow wire. (12 Volts)
Worried about shock?
Two D cell flashlight batteries store 3 Volts DC.
The dangerous voltages are contained in the metal case of the Power Supply.
Also following Anti-Static Precautions, remove the Ram Memory module/s ('Stick'), and clean the gold plated contact pins with an eraser.
Handle the Ram Memory module by the BODY.
The Body is everywhere on the ram memory module, Except the gold plated contact pins on the bottom,http://en.wikipedia.org/wiki/File:Generic_DDR_Memory_%28Xytram%29.jpg
The Body is coated with a see-through protective plastic.
If you are using an eraser on the end of a pencil, and not a solid eraser, refrain from letting the metal band of the pencil touch the gold plated contact pins.
Rub up, and down on EACH contact pin. BOTH sides. Doesn't take much pressure, and very much cleaning. The contact pins may not appear shiny, and bright when you are done. When you are finished it may seem like you have done nothing at all. I assure you that you have.
Doesn't take very much 'corrosion' to make a bad contact.
Use air to remove the eraser dust. If you have a can of compressed air you can use it. If not air pressure from your mouth will be sufficient.
Reinstall the ram memory module/s.
For additional questions please post in a Comment.