Some questions first:
1. Do you see ANYTHING when you power on your PC?
2. Does the power indicator light on the monitor change colors when you power on the PC (IE: Orange to blue)?
3. Do you see the POST screen or motherboard logo screen when PC is powered on?
4. Do the fans on the new graphics card spin up?
4. Can you include the make and model of your motherboard?
Not knowing any of the above, here is what I would do:
Troubleshooting 1: See if the monitor, not the GPU is the issue.
*If you can, try using your monitor with a different, known to be working computer.
Troubleshooting 1a:
*Many motherboards have built in GPU. Most of them that do, have a VGA port coming off the back of the motherboard; look at the rear of your PC and see if your PC has this port. The VGA port looks like this:
*if you see it, it means your motherboard has an on-board GPU. Sometimes, PC's can magically decide, on their own, to switch from using the PCI-E and decide to use the integrated graphics. This can especially happen when the dedicated PCI-E card is removed.
*With your graphics card uninstalled, plug your monitor into the VGA output that comes off your motherboard and see if you get an image on the monitor.
*If an image is displayed, go to your BIOS.
*There are different ways to access the BIOS on PC, most common being to press either the DEL key, F11 key or the F12 key while the computer is booting up.
*Once you are in BIOS, be sure that you have "use integrated GPU" or "integrated graphics accelerator" turned off. Sometimes, it will also have a setting for "Enable/use PCIE" if you see this, select "ON" or "Enable" or "1"
Troubleshooting step 2:
*If using an HDMI cable, try different HDMI cables. They are NOT all created equally, and as I found out, monitors can be picky about choosing which standard to work with.
***If any the above troubleshooting methods fail to produce an image on the monitor, especially plugging your monitor into another known working PC...Start thinking about the monitor, not the GPU, being the issue***
Troubleshooting step 3:
*This one is a long shot, but has solved A LOT of PC issues for me that, frankly, it had no business solving: SO, Turn off the power switch on your power supply, then unplug it. Take off the side panel, and locate the CMOS battery. The CMOS battery looks like this:
*With the CMOS battery removed, and the computer off, press the power button on your PC. Wait 3-5 minutes.
*Put CMOS battery back in, plug computer back in, turn power supply back on, try turning on computer.
Troubleshooting step 4:
*Most 580's have three different types of display ports. They look like this:
1 X DVI-I // DVI-D:
2 X Display ports:
And Finally, two HDMI ports, which I assume you know what that looks like, so I won't put a picture of those in here.
*Try using alternate ports if using HDMI or Display port, as there are two of each.
*Also, if using HDMI, try using the display ports, or if you are already using Display port, try using HDMI, or the DVI port (if your monitor accepts those types of connections)
Troubleshooting 5:
*Many motherboards come with multiple PCI-E slots. This is the slot which your new 580 plugs into, looks like this (the larger slots are for the graphics cards, ignore the smaller slots that are outlined:
*If you have access to your motherboard manual, be sure you are plugging the graphics card into the correct slot for a single card configuration. It is easy to uninstall the old graphics card and install the new one in a different slot, as they all look identical.
*If you cannot access the manual, you can browse to your manufacturer's website (IE: asus, gigabyte,MSI etc etc) and can usually locate a digital form of the owner's manual there.
*Typically, the top slot is the slot you would want to use for a single card configuration, but refer to your manufacturer's manual to be sure.
Troubleshooting 6
*500 watt power supply is pretty much the minimum recommended power rating for a 580. It could be that your power supply is not sufficient or efficient enough to run the 580.
Troubleshooting 7
*Ensure you have both the 8 and 6 pin power connectors plugged into the 580.
Troubleshooting 8
*Re-seat the 580 in the correct PCI-E slot.
Or, do none of that, and just claim it's DOA and RMA to whomever you bought it from and wait for a new one to show up 
Oh, the joys of owning/working on your own Computer
