I have a emachine T6420

This is eMachines Support, T6420 desktop computer>Specifications, and downloads page for the User Guides,

http://www.emachines.com/support/product_support.html?cat=Desktops&subcat=T%20Series&model=T6420

The eMachines T6420 does not come with a graphics card from the factory.
It comes with Integrated Graphics.

Integrated Graphics is also known as OnBoard Graphics.
On the motherBoard

The GPU is soldered to the motherboard.
(Graphics Processing Unit,

http://en.wikipedia.org/wiki/GPU

OR,
someone has installed a graphics card, into the PCI Express x16 slot on the motherboard.

You can determine real easily if your computer is using Integrated Graphics, or a graphics card.

If it is using Integrated Graphics, the monitor will be plugged into the I/O area, on the back of the computer, and into a VGA port,

http://en.wikipedia.org/wiki/VGA_connector

(I/O area;

I/O = Input/Output

The I/O area, is the area that has the rectangular metal shield, and Input/Output devices are plugged into this area.

Devices such as a Keyboard, Mouse, Printer, and so on )

HOWEVER, with the motherboard used in an eMachines T6420, the I/O area could also have a white DVI port.

http://en.wikipedia.org/wiki/Digital_Visual_Interface

This was an option offered. The DVI port still connects to the Integrated Graphics on the motherboard.

The eMachines T6420 uses an MSI MS-7207 motherboard.
(Micro Star International, Inc.)

http://www.tigerdirect.com/applications/searchtools/item-detailsInactive.asp?EdpNo=2665522

The above link is to,
1) Show the motherboard
2) Provides a link to the Motherboard Manual

Looking at the photo of the motherboard, you will see two long White slots.
Above these two slots you will see a longer White slot.

The bottom two white slots are PCI expansion slots.
The top longer slot is a PCI Express x16, expansion slot.

This is where a PCI Express graphics card would go.

I would like you to scroll to the bottom of the page, to the Product Resources heading.

Under it is a link to the Motherboard Manual, for the MSI MS-7207 motherboard.
Also known as K8NGM2-L

Left-click on the blue - MSI K8NGM2 Series Manual.

You will be looking at a blank white page, as the PDF file downloads in the background.
Took 34 seconds with a medium speed DSL connection, for me to download it.

NOW,
No Signal on the monitor, means the monitor is Not receiving a video signal.

1) Check the monitor cable if it is removable. Use it on another working computer to insure it is good.

2) If the monitor cable proves to be good, the next area to check would be the graphics.

If the computer is using Integrated Graphics, and no one has installed a graphics card, the test would be to install a graphics card, and plug the monitor into it.

Computer off, unplugged from power, Anti-Static precautions are observed before reaching inside the computer, and handling the graphics card.

I don't think the problem is the graphics. Could be wrong, but we'll see.
Let's carry the diagnosis past the graphics for now.

I think the problem could be due to a bad Power Supply.
Weak Voltage power rail.

Enough power to light lights, and spin fans (Perhaps), but not enough to turn the Processor on.
(Fans may be spinning, or may just spin a few times, and stop)

1) ALL the lights 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.

Continued in an additional Comment.

check plug for bent pins.....

You need to replace the fan, it will cause further damage if you don't.

I’m happy to assist further over the phone at https://www.6ya.com/expert/joe_8b8c2cd6ce148309

First of all plug another monitor to the VC on your PC and do the same for the SamSung"I over heat"SyncMaster2 053BW. Reply!

Change the order so that the CD boots first. Try to run Repair from the Windows CD. If unsuccessful, you'll need to reinstall Windows.

This problem comes due to problem with RAM and CMOS.

first you get RAM outside the CPU and clean properly by brush,and put into the RAM SLOT.

it work.
otherwise ,you have to change RAM.

Remove and reinstall memory.

No you cant, the gateway GX7018E uses integrated graphics like the T6420. You will have to buy a cheap PCI or PCI-E graphics card to test it out. 

Most likely your video card is broken. To check if the motherboard isnt broken either, I would suggest you test usb, sound, and other features of the motherboard.
 If your motherboard isnt broken then you have to buy a "pci-e graphics card" at your local store. Shouldnt cost much, about $40-70. Hope that helps.

That's exactly what it sounds like. I would put in a new video card (or used one, if you have one) into one of the expansion slots and run it from there.

I would suggest to go into VGA mode just to make sure, but since your monitor never even gets off standby, I'd be 99% it's the onboard video.

Hope this helps!