I had my video card replaced by Apple (a few years ago) in my 2011 iMac 27" (iMac12,2). The genius bar said there was a problem with video cards in 2011 iMac 27" models. I don't know what (or if) other models were affected.
My troubleshooting sequence would be, from the simple and most common issues to the more serious and involved:
Reset the PRAM --Always try this first if your computer acts up -- Turn off computer completely, hold down all four "cmd opt P R" keys while powering back on. Keep holding the keys down until you hear a "bong" sound. Usually I keep holding them down until I hear that sound 3 times before releasing.
To rule out software issues (corrupt OS or drivers), try booting from the original OS X DVD that came with the computer.
Connect a 2nd monitor to the Display Port and see if it works or not. If it still doesn't work, it's probably your video card. If it does work, there's something with the main monitor itself. In either case, it's time to make a decision to fix or replace. Both can be expensive, or not as much, depending on your experience fixing computers or have a friend that's good at fixing them.
The hardest part about this (besides parting with the cash to fix or replace) will be backing up the data on the hard drive. Hopefully you have a TimeMachine, iCloud or other backup.
Thanks everyone for your responses. The iMac was purchased from Apple in Feb. 2011, and is a mid 2010 21.5 Inch, original software was Mac OS X. The backlight power supply was replaced under extended warranty in May, 2012. If my wife (current user) has been upgrading software, it is probably running 10.14.2 Mojave. She is not sure if it is backed up, but I think it is on our Apple Cloud storage as we use the share plan. I am using my 21.5 2 year-old iMac to type this. We have tried the PRAM reset - no response. I will try rebooting with the original software.
According to my wife there was no event that caused it (except perhaps too much retail therapy). According to the local Apple authorized repair shop, parts for this model are no longer available. Is this accurate?
Should I take it to an Apple store when we go to Charlotte, NC next week? Any thoughts on a "refurbished"?
Is there a way for you to check whether or not that machine is backed up, on your shared Apple Cloud storage? I use my own "local" drives (actually SSDs) for backups, so I really cannot help with "Cloud" features.
As for refurbished, I prefer brand new myself. Our youngest son bought as a refurbished Philips 46" Plasma TV a few years ago, but it went bad after less than 3 years of use. On the other hand, I had previously purchased a refurbished power supply "brick" for an older Mac laptop I had, and it stood up well.
How much of a budget do you have to spend on a newer iMac?
To be honest, If it's just a computer to you, then I would use this opportunity to upgrade to a newer one. It's an old computer and fixing it is probably not worth the effort since repairs are usually not exactly inexpensive and you end up with an old computer. The used price for a working iMac 21.5-inch Mid 2010 computer is about $125.
Some places and tools I use when looking for used computers (it doesn't hurt to look):
Craigslist -- You can find good used computers on craigslist that are newer and better than what you have. You need to read the postings and look at the pics CAREFULLY and CRITICALLY. About half of what I find on craigslist are broken or there's something wrong. But I find most sellers are pretty honest and will answer questions you may have. If you see something in a picture that doesn't look right, ask them about it.
A great resource for used Mac specs and price information is the application Mactracker. It's free and you can download it from the App Store. Awesome. If you look up a computer and click on the average used price, it takes you to a great website that will look up used price trends from auction sites like eBay. If anything, It'll give you an idea of what something is worth.