If you want a cheap PC that's not junk, you're going to have to save up, and look at secondhand machines/parts.
For most games, you should be good if you buy an old Core2-based machine, and upgrade the video card to something a bit newer. Upgrading the ram is a good idea too, as you're going to want 4GB minimum for day-to-day tasks nowadays. If you're just going to be watching YouTube and going on Facebook, you could skip the video card upgrade altogether, though I'd at least recommend getting a $50 Newegg special, just to have something.
An AMD-based rig might be OK as well, but make sure it's something newer. Athlon 64 X2 machines are as common as dirt, but they're almost useless for gaming unless you have the patience of a saint, or overclocking skills. I overclocked an Athlon 64 x2 machine and installed a cheap video card from eBay, and it did a not-bad job for running lower-end games, but it was massively unstable too. It was an OEM board and I had to do all the overclocking from Windows, so I didn't have access to things like voltage controls (as far as I remember anyway). Anything with an Athlon II, Phenom II, AMD FX, or AMD A4/A6/A8 chip should be alright
As far as an i3/i5/i7 machine, that's almost out of the question, except if someone gives you one for free. My dad gave me his Core i5 Alienware when he was tired of it overheating, and it ran well for me for a while after I cleaned it, but the GPU eventually succumbed to the effects of having its life shortened by overheating.
Also, don't let someone con you into buying a Pentium 4, an Athlon XP, a Pentium D, or a single-core Athlon 64. These are all really old CPUs, and they won't deliver the performance you want for modern games, even with maxed ram and a good video card. Now, the Pentium D and the single-core Athlon 64 may be a bit debatable, but I'd still avoid them.
With Celerons, I would only buy one if it was Ivy Bridge-generation or newer, as none of the older ones were dual-cored, and you pretty much need a dual-core CPU nowadays. If the seller/store you're buying it from doesn't know if it's an Ivy Bridge or not, don't buy it.
But to sum it up, look for:
- Core 2-generation CPU or newer with at least two cores
- 4GB RAM minimum, 8GB is better but unlikely
- ATI or NVIDIA GPU with at least 1GB of RAM.
- (optional, depends on how courageous you are) a motherboard and CPU cooler capable of overclocking, so you can squeeze more out of your hardware
At this price point, it's not too worth getting worked up over other details. It is good however to get a new, unused hard drive and power supply, since these two components tend to wear out more quickly than other parts of a computer (except maybe the video card, since they aren't properly cooled half the time).