Yeah, knowing a location would help. In the US?
For that kind of money, assuming you're talking US online prices for the system only and not a monitor, I would try to look for something with a graphics card that offers at least GTX 1060 or RX 480 / 570 / 580 -level performance, and preferably a 6-core Ryzen or Intel CPU (though a recent quad-core might be okay). For gaming at 1080p resolution, you probably wouldn't want to cheap out any more on the graphics card. Systems with a decent graphics card and processor in this price range will most likely only have 8GB of RAM though, which isn't ideal for a few of the newest games, though nearly all games will still run rather well on 8GB for the time being. You might be in need of a RAM upgrade within the next couple years though.
Something like this (which comes up as "Amazon's Choice" when searching for "Gaming PC" there) would probably be decent enough, and you probably wouldn't currently be able to get much better performance out of a pre-built system for $800. It might be worth looking around in case any better options pop up though...
https://www.amazon.com/CYBERPOWERPC-GXIVR8020A5-Desktop-i5-8400-Processor/dp/B07B6H4GNY/
I have no real first-hand experience with CyberPowerPC systems, but they do seem to pack in reasonably decent levels of gaming performance for the money, at least for a pre-built system.